text:
\section{\bf Introduction}

In this article we continue the study of Ricci solitons using
 methods of dynamical systems, focusing on the
case of steady solitons. A Ricci soliton consists of
a complete Riemannian metric $g$ and a (complete) vector field $X$
satisfying the equation:
\begin{equation} \label{LieRS}
{\ric}(g) + \frac{1}{2} \,{\sf L}_{X} g + \frac{\epsilon}{2} \, g = 0
\end{equation}
where $\epsilon$ is a real constant and $\sf L$ denotes the Lie derivative.
The soliton is {\em steady} if $\epsilon =0$, {\em expanding}
if $\epsilon >0$ and {\em shrinking} if $\epsilon < 0$.
If $X$ is Killing, then $g$ is Einstein and the soliton is
called {\em trivial}. By the work of Perelman \cite{Per}, nontrivial steady and
expanding solitons must be noncompact.

A Ricci soliton is of {\em gradient type} if the vector field $X$ is the
gradient of a globally defined smooth function, referred to as the
{\em  soliton potential}. Many examples of K\"ahler gradient Ricci solitons
of all three types exist in the literature, see, for example, \cite{Ko},
\cite{Cao}, \cite{FIK}, \cite{WZh}, \cite{ACGT}, \cite{PS}, and  \cite{DW1}.
By contrast, fewer non-K\"ahler gradient Ricci solitons are known.

In \cite{DW2} a family of non-K\"ahler steady gradient Ricci solitons was
constructed generalising the rotationally symmetric Bryant solitons \cite{Bry} on
$\R^n \, (n > 2)$ and Ivey's related examples \cite{Iv}. The manifolds in this family
consisted of warped products on an arbitrary number of Einstein factors with positive
Einstein constants, and they exhibited asymptotically paraboloid geometry, like the
examples of Bryant and Ivey. In particular, they gave examples of steady
gradient Ricci solitons in dimensions greater than three which are not rotationally
symmetric. For dimension $3$, Brendle has recently proved that the Bryant soliton is the
only non-trivial $\kappa$-noncollapsed steady gradient Ricci soliton \cite{Br1}.

In this paper we use methods similar to those in \cite{DW2}, but allow one
of the factors in the warped product to be a circle (hence with zero Einstein
constant). We obtain complete steady Ricci solitons whose asymptotics are a mixture
of the paraboloid Bryant asymptotics and the cylindrical asymptotics
of Hamilton's cigar soliton \cite{Ha1}. More precisely, the metric
on the circle factor is asymptotically constant while those
on the other factors asymptotically grow like the geodesic distance
coordinate. This type of asymptotics has been observed previously for
K\"ahler steady solitons, cf \cite{DW1}. The general results of
Buzano \cite{Buz} now allow us to dispense with some of the analysis of
\cite{DW2} concerning smoothness at the other end of the manifold,
where the circle collapses. It should be mentioned that the special case
in which there are two factors (including the circle factor) was discussed
in Ivey's Duke thesis and in \cite{Iv}.

For dimensions greater than three, Brendle has obtained an analogue of his
$3$-dimensional rigidity theorem for complete steady gradient Ricci solitons \cite{Br2}.
Under the hypotheses of positivity of sectional curvatures and of being
``asymptotically cylindrical", he proves that such a soliton must be the
Bryant soliton. We note that our examples always have some negative
sectional curvatures when the number of positive Einstein factors
is at least $2$, although the Ricci tensor is non-negative. As for the
asymptotically cylindrical property, our examples do satisfy the upper
and lower scalar curvature bounds, but not the stronger requirement
involving the Gromov-Hausdorff convergence of rescaled flows to shrinking
cylinders.

In addition we prove some general results about steady solitons of cohomogeneity
one type, including monotonicity and concavity results for the soliton
potential, and decay estimates for the ambient scalar curvature and the
mean curvature of the hypersurfaces.

We also study a family of solutions to our equations that yield
complete Ricci-flat metrics. These are related to examples of
B\"ohm \cite{Bo1} which are multiple warped products whose factors are
Einstein metrics of positive scalar curvature. In our examples, as in the
soliton case above, one of these factors is replaced by a circle.
The resulting equations are rather different in character from those
considered by B\"ohm, due to the fact that we no longer have a solution
representing a cone over a positive scalar curvature Einstein metric on the
hypersurface (which acts as an attractor for the B\"ohm system). Note
that the special case in which there are only two factors is explicitly
integrable, and was discussed in \cite{BB} and \cite{Be}. This special case
includes the Riemannian Schwarzschild metric.

Combining our construction with the work of Boyer, Galicki and Koll{\'a}r
on Einstein metrics on exotic spheres,  the work of K. Kawakubo and R. Schultz,
and the recent work of Hill, Hopkins and Ravenel, we deduce that in
all dimensions congruent to $3$ mod $4$ other than $3, 7, 15, 31, 63$ and
possibly $127$, there are homeomorphic but not diffeomorphic complete
non-compact Ricci-flat manifolds as well as steady gradient Ricci solitons
(cf Corollary \ref{diffstr}).

The above examples all fall within the class of multiple warped products
on Einstein factors with nonnegative Einstein constant.
The analysis of the dynamical system in such cases is aided by the
fact that the scalar curvature of the hypersurface is bounded below.
In the examples treated in \cite{DW2} and (in the Einstein case) \cite{Bo1},
the scalar curvature is in fact strictly positive. This is related to the fact
that the Lyapunov function defined in \cite{DW2} for a general
cohomogeneity one steady soliton system is in these cases actually a positive
definite quadratic form (up to an additive constant). This gives coercive
estimates on the flow which facilitate the analysis. In the examples of the
present paper, where one factor in the warped product is flat, the
Lyapunov is no longer definite, but it becomes definite upon restriction
to a subsystem of one dimension lower, and this turns out to be enough
for many of the arguments to go through.

It is therefore of great interest to consider the soliton equation in cases
where the hypersurfaces have more complicated scalar curvature. One natural
class of examples are hypersurfaces which are the total spaces of Riemannian
submersions for which the hypersurface metric involves two functions, one
scaling the base and one the fibre of the submersion. If the fibre is a circle
this leads us to ans\"atze familiar in the Einstein case from the work of
Calabi and B\'erard Bergery. In the soliton case, K\"ahler examples are known
(see the references above), although the general case is still not well-understood.
For higher-dimensional fibres the equations are more complicated. In the Einstein case
B\"ohm obtained existence results for compact examples in some low dimensions \cite{Bo2}.

In the last section we discuss some results from a numerical study of the steady soliton
equation for these more complicated principal orbit types. Viewing the quaternionic
projective space as a quaternionic K\"ahler manifold, we take its twistor bundle
(resp. canonical ${\rm Sp}(1)$ bundle) as the principal orbits in the associated
$\R^3$ (resp. $\R^4$) bundle over $\HH\PP^m, m \geq 1$. We produce numerical evidence
of complete steady gradient Ricci soliton structures on these bundles. Note that
the existence of complete Ricci-flat metrics on these bundles, including ones
with special holonomy, was considered by \cite{BryS}, \cite{GPP} and \cite{Bo1}.
In the soliton case, however, we did not detect any difference between low and
high dimensional cases.

\section{\bf Generalities on cohomogeneity one steady solitons}

In \cite{DW1} two of the authors set up the formalism for Ricci solitons of cohomogeneity one.
More precisely, we considered the situation of a manifold $M$ with an open dense set
foliated by equidistant diffeomorphic hypersurfaces $P_t$ of real dimension $n$.
In other words, the metric is taken to be of the form $\bar{g}=dt^2 + g_t$ where $g_t$
is a metric on $P_t$ and $t$ is the arclength coordinate along a geodesic orthogonal
to the hypersurfaces. This formalism is somewhat more general than the cohomogeneity one
ansatz, as it allows us to consider metrics with little or no symmetry provided that
appropriate additional conditions on $P_t$ are satisfied, see the following as well as
Remarks 2.18 and 3.18 in \cite{DW1}.

We shall consider solitons of {\em gradient type}, that is, we take $X = {\rm grad} \; u$
for a function $u$. Equation (\ref{LieRS}) then becomes
\begin{equation} \label{gradRS}
{\ric}(\bar{g}) + {\rm Hess}(u) + \frac{\epsilon}{2} \, \bar{g} = 0.
\end{equation}
We will further suppose, in line with the cohomogeneity one formalism, that
$u$ is a function of $t$ only, and treat it as both a smooth function on
the manifold and a function of the single variable $t$.

We let $r_t$ denote the Ricci tensor of $g_t$, viewed as an endomorphism via $g_t$.
Then we can define $L_t$, the shape operator of the hypersurfaces, by the equation
$\dot{g_t} = 2 g_t L_t$. We assume that the scalar curvature $S_t=\tr(r_t)$ and the
mean curvature $\tr(L_t)$ (with respect to the normal $\nu=\frac{\partial}{\partial t}$)
are constant on each hypersurface. We shall often in the future suppress the $t$-dependence
in the above tensors.

In this setting, the above equation becomes the system (cf \S 1 of \cite{DW1})
\begin{eqnarray}
  -{\rm tr} (\dot{L}) - {\rm tr} (L^2) + \ddot{u} + \frac{\epsilon}{2} &=& 0,  \label{TT} \\
    r - ({\rm tr}\, L)L - \dot{L} + \dot{u} L + \frac{\epsilon}{2} \, \I &=& 0, \label{SS}\\
d ({\rm tr} L) + \delta^{\nabla} L &=& 0.  \label{TS}
\end{eqnarray}
The first two equations represent the components of the equation
in the $\frac{\partial}{\partial t}$ direction and in the directions
tangent to $P$, respectively. Also, $\delta^{\nabla} L$ denotes the
codifferential for $TP$-valued $1$-forms, and the third equation represents
the equation in mixed directions.

The above assumptions are satisfied, for example, if $M$ is
of cohomogeneity one with respect to an isometric Lie group action.
They are satisfied also when $M$ is a multiple warped product over an
interval, which is the situation we focus on in this paper.

In the warped product case the final equation involving the
codifferential automatically holds. This is also true for
cohomogeneity one metrics that are {\em monotypic}, i.e. when there
are no repeated irreducible summands in the isotropy representation
of the principal orbits (cf \cite{BB}, Prop. 3.18).

We have a conservation law
\begin{equation} \label{cons1}
\ddot{u} + (-\dot{u} + {\rm tr}\, L) \dot{u}  -\epsilon u = C
\end{equation}
for some constant $C$.
Using the equations this may be rewritten as
\begin{equation}
{\rm tr} (r_t) + {\rm tr} (L^2) - (\dot{u} - {\rm tr}\, L )^2 - \epsilon u +
\frac{1}{2}(n-1) \epsilon = C.
\end{equation}

The term ${\rm tr} (r_t)$ is the scalar curvature $S$ of the principal
orbits. Recall that if $\bar{R}$ denotes the scalar curvature of the
ambient metric, from now on written as $\bar{g} = dt^2 + g_t$, then
$$ \bar{R} = -2 {\rm tr} (\dot{L}) - {\rm tr} (L^2) - ({\rm tr} L)^2 +S. $$
We can deduce the equality
\begin{equation}\label{ham}
\bar{R} + \dot{u}^2 + \epsilon u = -C -\frac{\epsilon}{2}(n+1),
\end{equation}
which is just the cohomogeneity one version of Hamilton's
identity $\bar{R} + |\nabla u|^2 + \epsilon u =\;$ constant.

\smallskip

We now specialise to the case of {\em steady solitons}, that is, $\epsilon =0$.
The conservation law is now
\begin{equation} \label{cons2}
{\rm tr} (r_t) + {\rm tr} (L^2) - (\dot{u} - {\rm tr}\,  L)^2 = C.
\end{equation}

In \cite{DW2} the following result was proved.
\begin{prop}
The function $(\dot{u} - {\rm tr}\, L)^{-2}$
is a Lyapunov function, that is, it is monotonic on each interval on which it is defined.
\end{prop}

\medskip
The expression $-\dot{u} + {\rm tr} \, L$ is the soliton version of the
hypersurface mean curvature. It occurs so frequently in our analysis
that we shall introduce some special notation for it
\[
\xi := -\dot{u} + {\rm tr} \, L.
\]

\begin{rmk}
The conservation law (\ref{cons2}) shows that our Lyapunov function is a constant multiple
of
\[
\frac{{\rm tr}(r_t) + {\rm tr} (L^2)}{(\dot{u} - {\rm tr}\; L)^2}  -1.
\]
\end{rmk}
It is convenient to
define a function ${\mathcal L } = \frac{C}{\xi^2}$ and we refer to this
as {\em the} Lyapunov.

It is often useful to define a new independent variable $s$ by
\begin{equation} \label{st}
\frac{d}{ds} := \frac{1}{\xi} \frac{d}{dt} =
\sqrt{\frac{\mathcal L}{C}} \frac{d}{dt}
\end{equation}
and use a prime to denote $\frac{d}{ds}$. We shall see presently that $\xi$ is always
positive in the case of a complete steady soliton.
 Another useful quantity is
\[
{\mathcal H} = \frac{{\rm tr} L}{\xi} = 1 + \frac{\dot{u}}{\xi}
= 1 +u^\prime,
\]
which was introduced in \cite{DW3}, \cite{DHW}.

In the steady case the locus $\{ {\mathcal L} =0, {\mathcal H}=1 \}$
is invariant under the flow, and trajectories in this region of
phase space correspond to trivial solitons, i.e., ones where
$u$ is constant and $g$ is an Einstein metric. Analogous
statements hold in the expanding and shrinking cases if we modify
$\mathcal L$ appropriately. We refer the reader to \cite{DW3}
and \cite{DHW} for further discussion.

\bigskip

We now describe some general results about complete cohomogeneity one steady
solitons of gradient type. Some of these results can be deduced from theorems
about general solitons found in e.g., \cite{FR}, \cite{MS}, \cite{Wu}, \cite{WeiWu}.
However, in the cohomogeneity one situation, the statements sometimes take on a
stronger or more precise form, which will be useful for checking asymptotic
behaviour in numerical studies. We have also included their proofs here. Besides
being more elementary, they involve ideas which are useful for analysing
existence questions in the cohomogeneity one case.

We shall be looking at complete noncompact steady solitons with one special
orbit, in which case  we may assume, without loss of generality, that $t \in [0, \infty)$
and the special orbit occurs at $t=0$. We let $k$ denote the dimension of the
collapsing sphere at $t=0$.

The equation (\ref{ham}) becomes
\begin{equation} \label{ham2}
\bar{R} + (\dot{u})^2 = -C
\end{equation}
for complete steady solitons. We recall the result of B. L. Chen \cite{Chen}
that $\bar{R} \geq 0$ for complete steady solitons, with equality iff $\bar{g}$
is Ricci-flat. Hence we deduce the important inequality
\[
C < 0
\]
for non-trivial steady solitons.
Note that this is a global consequence of completeness which does not follow
from examining local existence in some neighbourhood of a singular orbit
(cf. \cite{Buz}). The specific value of $C$ is unimportant as it can be changed
by a positive multiple via a homothety of the soliton metric.

\begin{prop} \label{steady-pot}
The soliton potential is strictly decreasing and strictly concave
on $(0, \infty)$.
\end{prop}

\begin{proof}
Let $t_0$ be a critical point of $u$ in $(0, \infty)$. The
conservation law (\ref{cons1}) with $\epsilon =0$, together with the
negativity of $C$, show that $u$ is strictly concave in a neighbourhood of $t_0$.
So the critical points of $u$ are isolated and nondegenerate.

Next let $t_0 < t_1$ be two consecutive critical points. The concavity
statement means there must be a critical point between $t_0$ and $t_1$, a
contradiction. So if a critical point exists it is unique.

The smoothness conditions at the special orbit imply
$\dot{u}(0)=0$ and $\xi = \frac{k}{t} + O(t)$ near $t=0$. Substituting into
(\ref{cons1}) yields $(k+1) \ddot{u}(0) = C<0$ so in fact we
have concavity at $t=0$ also.
Hence the above argument shows there are in fact no critical points
of $u$ in $(0, \infty)$ and $u$ is hence strictly decreasing.

Now set $y = \dot{u}$ and differentiate (\ref{cons1}); using (\ref{TT})
we obtain
\[
\ddot{y} + \xi \dot{y} - {\rm tr}(L^2) y =0.
\]
We know $y \leq 0$ on $[0, \infty)$ with equality only at $0$. Also
$\dot{y}(0)<0$ and $y^2 < -C$. If $t_0 >0$ is a critical point of $y$, then
$\ddot{y}(t_0) \leq 0$ with equality only if $L(t_0)=0$. But this
implies, by (\ref{cons1}), that $-\dot{u}(t_0)^2 =C$, which contradicts
positivity of $\bar{R}$. Hence $y$ is strictly concave at its critical points.
Looking at the first critical point and using the above information on $y$
now shows no critical points can exist. So $\ddot{u} = \dot{y}$ is negative
for all  $t>0$.
\end{proof}

\begin{prop} \label{steady-MC}
The mean curvature ${\rm tr} \; L$ is strictly decreasing
and satisfies $0 < {\rm tr} \; L \leq \frac{n}{t}$. The generalised
mean curvature $\xi$ is strictly decreasing and tends to $\sqrt{-C}$
as $t$ tends to $\infty$.
\end{prop}

\begin{proof}
The preceding proposition shows that $\frac{d}{dt}({\rm tr} \; L) =
-{\rm tr} (L^2) + \ddot{u}$ is negative. By Cauchy-Schwartz, we have
$\frac{d}{dt} ({\rm tr} \; L) < -\frac{1}{n}( {\rm tr} \; L)^2$.

Suppose ${\rm tr} \; L$ vanishes at $t_0$. Then ${\rm tr} \; L < 0$ on
$(t_0, \infty)$.
Integrating the inequality $\frac{d}{dt}({\rm tr} \; L) \leq \ddot{u}$
from $t_0 + \delta$ to $t$, where $\delta>0$, yields
\[
\frac{\dot{v}}{v} = ({\rm tr}\; L)(t) \leq ({\rm tr}\; L)(t_0 + \delta)
+ \dot{u}(t) - \dot{u}(t_0 + \delta) < \dot{u}(t) - \dot{u}(t_0 + \delta)
\]
where $v(t)$ is the volume of the metric $g_t$ relative to a fixed
invariant background metric on the principal orbit.
Since $\dot{u}$ is strictly decreasing, and bounded by (\ref{ham2}),
it tends to a negative constant $-a$ as $t$ tends to $\infty$. For sufficiently
large $t$ we may assume that
\[
\dot{u}(t) + a < \frac{1}{2}( \dot{u}(t_0 + \delta) +a ) := \alpha
\]
So $\frac{\dot{v}}{v}$ is less than the negative constant $-\alpha$, which
implies the metric has finite volume, a contradiction to Theorem 1.11 in
\cite{MS}.

It follows that $\tr \; L$ never vanishes and hence is positive
everywhere, since it tends to $+ \infty$ at $t=0$. Now for $0 < \eta < T$
we have
\[
\int_{t= \eta}^{t= T}  \frac{d({\rm tr}\; L)}{({\rm tr}\; L)^2}
< -\frac{1}{n} \int_{t =\eta}^{t = T} dt.
\]
Therefore
\[
\frac{1}{({\rm tr}\; L)(\eta)} - \frac{1}{({\rm tr}\; L)(T)}
< -\frac{1}{n} (T - \eta)
\]
which, on letting $\eta \rightarrow 0,$ gives us our claimed upper bound
on ${\rm tr} \; L$.

As $\dot{u}$ is bounded below and decreasing,
$\lim_{t \rightarrow \infty} \dot{u}\, {\rm tr} \, L =0$. Now the conservation law
(\ref{cons1}) implies that $\lim_{t \rightarrow \infty} \ddot{u}$ exists. Since
$\ddot{u} < 0$,  the boundedness of $\dot{u}$ implies that
 $\lim_{t \rightarrow \infty} \ddot{u} = 0$.
The conservation law then yields $a = \sqrt{-C}$.

Finally, by Eq. (\ref{TT}), $\dot{\xi} = -\tr (L^2) =-\tr ((L^{(0)})^2) -\frac{1}{n} (\tr \,L)^2 < 0$
since we have shown that $\tr \;L > 0$. The limiting value of $\xi$ is then that of $-\dot{u},$
i.e., $\sqrt{-C},$ as $\tr \;L$ tends to $0$.
\end{proof}

\begin{rmk}
We have shown for complete steady gradient Ricci solitons of cohomogeneity one
with a special orbit at one end that $\dot{u}$ tends to a negative
constant and $\ddot{u}$ tends to $0$ as $t$ tends to $\infty$. So the soliton
potential $u$ will have asymptotically linear behaviour. In numerical searches
it is of course not possible to generate solutions over an infinite interval,
so the asymptotic behaviour of quantities such as the soliton potential provides
a valuable check that a soliton has in fact been found numerically. We shall consider
numerical examples in \S 5.
\end{rmk}

\begin{cor}
The ambient scalar curvature $\bar{R}$ is decreasing and tends to zero
as $t$ tends to $\infty$. Furthermore, we have
$$ 0 < -\dot{u}\,\tr \,L < \bar{R} < 2\sqrt{-C}\left(\frac{n}{t}\right) + \frac{n^2}{t^2}. $$
\end{cor}
\begin{proof}
By (\ref{ham2}), $\frac{d}{dt} \bar{R} = -2 \dot{u} \ddot{u} <0$.
The limiting value follows from the above proposition and (\ref{ham2}).

Next, using (\ref{cons1}) followed by (\ref{ham2}), we obtain
$\frac{d}{dt} \bar{R} = 2 \dot{u} (\bar{R} +  \dot{u}\, \tr \, L). $
Since $\frac{d}{dt} \bar{R} < 0$ and $\dot{u} < 0$, we deduce the lower bound.
For the upper bound, note that by the conservation law (\ref{cons2}) and
Proposition \ref{steady-MC}, we have $S + \tr(L^2) = \xi^2 +C > 0$.  On the
other hand, from the trace of Eq. (\ref{SS}) and  (\ref{cons2}) we have
$$S + \tr(L^2) = -\bar{R} + (\tr \, L)^2 - 2\dot{u}\, \tr\, L.$$
Therefore, using Proposition \ref{steady-MC} again, we deduce that
$$ \bar{R} < (\tr \,L)(\tr \,L - 2\dot{u}) \leq \frac{n}{t} \left(\frac{n}{t} + 2 \sqrt{-C}\right).$$
\end{proof}

\begin{rmk}
Note that from the limiting values of $\dot{u}$ and $\ddot{u}$ we also get
$\lim_{t \rightarrow \infty} \frac{d}{dt} \bar{R} = 0.$
The asymptotic behaviour of $\bar{R}$ for general steady solitons is given by
Theorem 3.4 in \cite{FR}, from which the asymptotic value of $\bar{R}$ also follows.
The upper bound for $\bar{R}$ above is somewhat stronger than what can be deduced from
the general upper bound given in Corollary 1.3 of \cite{Wu}. The upper bound
shows that for $t \geq 1$, $\bar{R} < (2\sqrt{-C}+n)\,\frac{n}{t}$, which is
independent of the principal orbit. It is unclear whether/when $\tr \; L$ has an
asymptotic lower bound of the form $\frac{\rm const}{t}$. This is an interesting
question, however, in view of the hypotheses in Brendle's rigidity result \cite{Br2}.
\end{rmk}

Finally, we discuss another Lyapunov function, a modification of which will play
an important role in \S 4. This function, denoted by ${\mathscr F}_0$ below, was first
considered  by C. B\"ohm in \cite{Bo1} for the Einstein case and was subsequently
studied in \cite{DHW} for the soliton case.

\begin{cor} \label{steady-F}
Let ${\mathscr F}_0$ denote the function
$v^{\frac{2}{n}}\left( S + \tr( (L^{(0)})^2 ) \right)$ defined on the velocity phase space
of the cohomogeneity one gradient Ricci soliton equations, where $L^{(0)}$ is the trace-free part of $L$.
Then ${\mathscr F}_0$ is non-increasing along the trajectory of a complete non-trivial steady soliton.
Furthermore, the function ${\mathcal F} :=v^{\frac{2}{n}}\left( S + \tr( L^2 ) \right)$ is strictly
decreasing along such a trajectory.
\end{cor}
\begin{proof} By Proposition 2.17 in \cite{DHW}, we have the formula
\begin{equation} \label{Fdot}
 \dot{\mathscr F}_0 = -2 v^{\frac{2}{n}} \ \tr((L^{(0)})^2) \left(\xi - \frac{1}{n} \tr L \right).
\end{equation}
However, by Propositions \ref{steady-pot} and \ref{steady-MC},
$$ \xi - \frac{1}{n} \tr \,L = -\dot{u} + \left(1 - \frac{1}{n} \right) \tr \, L > 0.$$
So ${\mathscr F}_0$ is strictly decreasing except where $L$ is a multiple of the identity.

As for the second statement, since $\tr( L^2) = \tr( (L^{(0)})^2 ) + \frac{1}{n} (\tr L)^2$,
it suffices to examine
\begin{eqnarray*}
      \frac{d}{dt} \left( v^{\frac{2}{n}} (\tr L)^2 \right) & = & \left(\frac{2}{n}\right) v^{\frac{2}{n}} (\tr L)^3
               + 2 v^{\frac{2}{n}} (\tr L)(\tr L)^{\cdot} \\
           & = & 2 v^{\frac{2}{n}} (\tr L) \left( \frac{1}{n}(\tr L)^2 - \tr(L^2) + \ddot{u} \right) \\
           & \leq & 2 v^{\frac{2}{n}} (\tr L)\, \ddot{u},
\end{eqnarray*}
where we have used (\ref{TT}) and the Cauchy-Schwartz inequality. By Propositions \ref{steady-pot}
and \ref{steady-MC} the last quantity is negative along the trajectory.
\end{proof}

\section{\bf Multiple warped products}
We now specialise to multiple warped products, that is
metrics of the form
\begin{equation} \label{metric}
dt^2 + \sum_{i=1}^{r} g_i^2(t)\,  h_i
\end{equation}
on $I \times M_1 \times \cdots \times M_r$\, where $I$ is an interval in
$\mathbb R,$ $r \geq 2,$ and $(M_i, h_i)$ are Einstein manifolds
with real dimensions $d_i$ and Einstein constants $\lambda_i$. Note that
$n=\sum_i d_i \geq 3$ once some $M_i$ is non-flat.

The shape operator and Ricci endomorphism are now given by
\begin{eqnarray*}
L &=& {\rm diag} \left( \frac{\dot{g_1}}{g_1} \,\I_{d_1}, \cdots,\frac{\dot{g_r}}{g_r} \,\I_{d_r}\right)  \\
r &=& {\rm diag} \left( \frac{\lambda_1}{g_1^2} \, \I_{d_1}, \cdots,\frac{\lambda_r}{g_r^2}\, \I_{d_r}\right)
\end{eqnarray*}
where $\I_m$ denotes the identity matrix of size $m$.
As in \cite{DHW}, we work with the variables
\begin{eqnarray}
X_i &=& \frac{\sqrt{d_i}}{\xi} \frac{\dot{g_i}}{g_i} \label{def-Xi}\\
Y_i &=& \frac{\sqrt{d_i}}{\xi} \frac{1}{ g_i} \label{def-Yi}
\end{eqnarray}
for $i=1, \ldots, r$.
Notice that the definition of $Y_i$ in \cite{DW2} and \cite{DW3} differs from ours
by a scale factor of $\sqrt{\lambda_i}$; the new choice is more appropriate
to our current situation where one of the $\lambda_i$ may be zero.
Now
\[
\sum_{j=1}^{r} \, X_j^2  = \frac{{\rm tr} (L^2)}{\xi^2} \quad\textup{and}\quad
\sum_{j=1}^{r} \, \lambda_j Y_j^2 = \frac{ {\rm tr} (r_t)}{\xi^2}.
\]
So the Lyapunov function becomes
\begin{equation} \label{Lyap}
{\mathcal L} := \frac{C}{\xi^2} = \sum_{i=1}^{r} \, (X_i^2 + \lambda_i Y_i^2)  - 1
\end{equation}
where $C$ is a nonzero constant.

As mentioned above, we introduce the new coordinate $s$ defined by
(\ref{st}) and use
 a prime ${ }^{\prime}$ to denote differentiation with respect to $s$.

In our new variables the Ricci soliton system (\ref{TT})-(\ref{SS}) with $\epsilon=0$
becomes
\begin{eqnarray}
X_{i}^{\prime} &=& X_i \left( \sum_{j=1}^{r} X_j^2 -1 \right) +
\frac{\lambda_i Y_i^2}{\sqrt{d_i}} \, , \label{eqnX} \\
Y_{i}^{\prime} &=&  Y_i \left( \sum_{j=1}^{r} X_j^2 -\frac{X_i}{\sqrt{d_i}}
\right) \label{eqnY}
\end{eqnarray}
for $i=1, \ldots, r$. Note that homothetic solutions of the system (\ref{TT})-(\ref{TS})
give rise to the same solution of the above system.

We shall be concerned exclusively with the multiple warped situation
for the rest of the paper. Recall that in this case equation (\ref{TS})
is automatically satisfied. Note also that the above equations imply the equation
\begin{equation} \label{eqnLyap}
{\mathcal L}^{\prime} = 2 {\mathcal L} \left( \sum_{i=1}^{r} X_i^2 \right),
\end{equation}
so ${\mathcal L} =0$ is flow-invariant.
We also use the notation ${\mathcal G} := \sum_{i=1}^{r} X_i^2$ as this
quantity often occurs in our calculations.

The quantity ${\mathcal H} = \frac{{\rm tr} L}{\xi}$ becomes
$\sum_{i=1}^{r} \sqrt{d_i} X_i$ in our new variables.
We have the equation
\[
({\mathcal H} - 1)^\prime = ({\mathcal H} -1 )({\mathcal G}-1) + \mathcal L
\]
so, as mentioned above, we see that the region
$\{ {\mathcal L} =0, {\mathcal H} =1 \}$ of phase space
corresponding to Ricci-flat metrics is flow-invariant.

While in \cite{DW2} all $\lambda_i$ were taken to be positive, that is, the
Einstein constants on each $M_i$ were positive, we shall now look at the
case where the collapsing factor $M_1$ is $S^1$, so $d_1=1$, $\lambda_1=0,$
and the remaining $\lambda_i$ are positive. Then the equation for $X_1$ becomes:
\[
X_{1}^{\prime} = X_1 \left( \sum_{j=1}^{r} X_j^2 -1 \right).
\]
Note in particular this means the locus $X_1 =0$ is flow-invariant.

Conversely, if we have a solution of the above system
(\ref{eqnX}), (\ref{eqnY}) with $\lambda_1=0$, in the
region ${\mathcal L} <0$ (so $C<0$), we may recover $t$ and the
metric components $g_i$ from
\begin{equation} \label{def-tgi}
dt = \sqrt{\frac{\mathcal L}{C}}\,\, ds, \ \ \ \ \
g_i = \frac{\sqrt{d_i}}{Y_i} \, \sqrt{\frac{\mathcal L}{C}}\,\,.
\end{equation}
We can choose $t=0$ to correspond to $s = -\infty$.

The soliton potential is recovered from integrating
\begin{equation} \label{def-u}
 \dot{u} = \tr(L) - \sqrt{\frac{C}{\mathcal L}} =
\sqrt{\frac{C}{\mathcal L}}\, ({\mathcal H} - 1),
\end{equation}
and $\tr(L)$ is calculated using
\begin{equation} \label{logdiff-gi}
 \frac{\dot{g_i}}{g_i} = \sqrt{\frac{C}{\mathcal L}} \frac{X_i}{\sqrt{d_i}}\, .
\end{equation}

The following lemma is a routine calculation.
\begin{lemma} \label{zeros}
Let $d_1 =1$ and $d_i > 1$ for $i > 1$, so that
$\lambda_i =0$ iff $i=1$. The stationary points of $($\ref{eqnX}$)$, $($\ref{eqnY}$)$
are now:
\begin{enumerate}
\item[$($i$)$] the origin

\item[$($ii$)$] points with $Y_i=0$ for all $i$, and $\sum_{i=1}^{r} X_i^2 =1$

\item[$($iii$)$] points given by
\[
X_i = \sqrt{d_i}\, \rho_A \;\; : \;\; Y_i^2 = \frac{d_i}{\lambda_i}\,\, \rho_A (1- \rho_A),
\;\;\;  i \in A
\]
and $X_i = Y_i =0$ for $i \notin A$, where $A$ is any nonempty subset
of $\{2, \ldots, r \}$, and $\rho_A = \left( \sum_{j \in A} d_j \right)^{-1}$

\item[$($iv$)$] the line where $X_i=0$ for all $i$ and $Y_i=0$ for $i >1$

\item[$($v$)$] the line where $X_1 =1$ and $X_i, Y_i =0$ for $i > 1$.

\end{enumerate}
\end{lemma}

Note that $\mathcal L$ equals $-1$ in case (i) and (iv), and equals $0$ in case
(ii), (iii) and (v).   Cases (iv) and (v) are special to the case $d_1=1$
and mean that in this situation the origin is a non-isolated critical
point.

\section{\bf Soliton solutions}

As in \cite{DW2}, we shall construct complete non-compact steady
soliton metrics where one factor $M_1$ collapses at one end,
corresponding to $t=0$.  For the collapse to be smooth we take
$M_1$ to be a sphere $S^{d_1}$. (Note that $d_1$ is the same as the dimension $k$
in \S 1.) The manifold underlying the Ricci soliton is then the total space of
a trivial vector bundle of rank $d_1 + 1$ over
$M_2 \times \cdots \times M_r$.  In our case, of course, $d_1=1$.
The initial conditions for the soliton solution
to be $C^2$ are the existence of the following limits:
\begin{equation} \label{bdy0}
g_1(0)=0   \;\; : \;\; g_i(0) = l_i \neq 0 \; (i > 1),
\end{equation}

\begin{equation} \label{bdy1}
\dot{g_1}(0)=1 \;\; : \;\; \dot{g_i}(0) = 0 \; (i > 1),
\end{equation}

\begin{equation} \label{bdy2}
\ddot{g}_{1}(0) =0 \;\; : \;\; \ddot{g_i}(0) \;\, {\rm  finite} \;
(i > 1),
\end{equation}

\begin{equation} \label{bdyu}
u(0) \, \,  {\rm finite}: \, \, \dot{u}(0)=0 \, \, : \,\ddot{u}(0) \, \, {\rm \, finite}.
\end{equation}

In our $X_i, Y_i$ variables, this means we consider trajectories in
the unstable manifold of the critical point $P_0$ of (\ref{eqnX})
and (\ref{eqnY}) given by
\[
X_1 = 1, \;\; Y_1 = 1, \;\; X_i =Y_i=0  \,\, (i > 1).
\]
This critical point lies on the level set ${\mathcal L}=0$ of the Lyapunov.

The linearisation about this critical point is the system
\begin{eqnarray*}
x_1^{\prime} &=& 2x_1 \\
y_1^\prime &=& x_1 \\
x_i^\prime &=& 0 \;\;\; (i \geq 2) \\
y_i^\prime &=& y_i \;\; (i \geq 2)
\end{eqnarray*}
with eigenvalues
$2$,  $1$ ($r-1$ times), and $0$ ($r$ times).

\medskip
In contrast to the situation of \cite{DW2} we now have a centre manifold.

\medskip
The results of \cite{Buz} now show we have an $(r-1)$-parameter family
of trajectories $\gamma(s)$ such that $\lim_{s \rightarrow -\infty} \gamma(s)
= P_0$ and pointing into the region ${\mathcal L} < 0$. As in
\cite{DW2}, (\ref{eqnLyap}) shows that such trajectories stay in
${\mathcal L} < 0$. We can moreover choose the trajectories to have $Y_i >0$
for all time (note that the locus $Y_i =0$ is always invariant under the flow).

Because $d_1=1$ and hence $\lambda_1 =0$, the Lyapunov
${\mathcal L} = \sum_{i=1}^{r} X_i^2  + \sum_{i=2}^{r} \lambda_i Y_i^2$
does not involve $Y_1$, so the region ${\mathcal L} \leq 0$ is no
longer compact, in contrast to the situation in \cite{DW2}.

However, since $\lambda_1=0$, the variable $Y_1$ only enters
into the equations through the equation for $Y_1^\prime$. Hence
by omitting (\ref{eqnY}) for $i=1$ we obtain a subsystem
of (\ref{eqnX}), (\ref{eqnY}) for $X_i \,(i=1, \ldots, r)$ and
$Y_i \,(i=2, \ldots, r)$ and on this new space, ${\mathcal L} \leq 0$
is compact.  Moreover, once we have a solution to the subsystem
we can recover $Y_1$ via
\[
Y_{1}(s) = Y_{1}(s_0) \exp \left(\int_{s_0}^{s} \sum_{j=1}^{r} X_j^2 -X_1
\right).
\]
The critical points for the subsystem are given by cases (i), (ii)
and (iii) of the critical points for the full system.
In particular $P_0$ corresponds to the critical point $\hat{P}_0 =
(1,0,\ldots, 0)$ in the subsystem, and we have an $r-1$ parameter
family of solutions emanating from this point and lying in the
region $Y_i > 0$ and ${\mathcal L} < 0$.

\medskip
Let us now analyse these trajectories in ${\mathcal L} < 0$.
For the subsystem, where this region is precompact, all
the variables are bounded by $1$ and the flow exists for all $s$.
Hence this is true for the original flow also.

The arguments of Prop 3.7 of \cite{DW2} show that the flow in the subsystem
converges to the origin, so ${\mathcal L}$ converges to $-1$. We deduce
from (\ref{st}) that as $s$ tends to $\infty$, so does $t$, hence the metric
is complete.

The proof of Lemma 4.4 (i) in \cite{DW2} carries over to show that
all $X_i$ are positive on the trajectory.
Using the arguments of Lemma 3.8 in \cite{DW2} we can show, using the equation
\[
\left(\frac{X_i}{Y_i^2} \right)^\prime = \left(\frac{X_i}{Y_i^2} \right)
\left(-1 - {\mathcal G} + \frac{2X_i}{\sqrt{d_i}} \right) + \frac{\lambda_i}{\sqrt{d_i}}\;,
\]
the following result.
\begin{lemma}
We have $\lim_{s \rightarrow \infty} \frac{X_i}{Y_i^2} = \frac{\lambda_i}{\sqrt{d_i}}$ for $i \geq 2$. \qed
\end{lemma}
(Recall that the $Y_i$ in the current paper differ from those in
\cite{DW2} by a $\sqrt{\lambda_i}$ scale factor.)

\medskip

Now
\[
\frac{1}{2} \frac{d}{dt} (g_i^2) = g_i \dot{g_i} = \frac{d_i}{Y_i^2}
\frac{\mathcal L}{C} \frac{X_i}{\sqrt{d_i}}
\sqrt{\frac{C}{\mathcal L} } =
\frac{\sqrt{d_i} X_i}{Y_i^2} \sqrt{ \frac{\mathcal L}{C}}
\rightarrow \frac{\lambda_i}{\sqrt{|C|}}
\]
as $s$ tends to $\infty$. Hence we deduce that as $t$ tends to $\infty$,
$g_i^2$ to leading order asymptotically behaves like
$ \frac{2 \lambda_i t}{\sqrt{-C}} $ for $i > 1$.

For $i=1$, we again use the equation
\[
\left(\frac{X_1}{Y_1^2} \right)^\prime = \left(\frac{X_1}{Y_1^2} \right)
\left(-1 - {\mathcal G} + 2X_1 \right),
\]
which we may write in the form
\[
(\log \psi)^{\prime} = - 1 + \phi
\]
where $\psi = \frac{X_1}{Y_1^2}$ and $\phi$ tends to $0$ as $s$ tends
to $\infty$. Choosing $0< \delta <1$ and $s_0$ such that $|\phi(s)| < \delta$ for
$s > s_0$, we integrate and obtain
\[
-(1 + \delta)(s - s_0) < \log \frac{\psi(s)}{\psi(s_0)} <
(-1 + \delta)(s-s_0)
\]
for $s > s_0$. Exponentiating gives
\[
e^{-(1 + \delta)(s-s_0)} < \frac{\psi(s)}{\psi(s_0)}
< e^{-(1 - \delta)(s-s_0)}
\]
so $\psi = \frac{X_1}{Y_1^2}$
decays to zero as $s \rightarrow \infty$.
Now, as in the $i>1$ case, we have
\[
\frac{1}{2} \frac{d}{dt} (g_1^2) \; dt=
\frac{X_1}{Y_1^2} \sqrt{ \frac{\mathcal L}{C}} \; dt =
\frac{X_1}{Y_1^2}  \frac{\mathcal L}{C} \; ds,
\]
and this integrand is positive and dominated by
$\frac{1}{|C|} \frac{X_1}{Y_1^2}.$ Integrating
and using the exponential bound above shows that the
increasing function $g_1^2$ is bounded above, hence converges
to a positive limit ${\alpha}^2$.

\begin{rmk} \label{Y1value}
Since $g_1(t)$ tends to $\alpha$ and $\xi$ tends to $\sqrt{-C}$ as $t$ tends
to infinity, it follows from (\ref{def-Yi}) that $Y_1$ tends to a positive
constant as $s$ tends to infinity. This means that the soliton trajectory
in the full $X_i, Y_i$ space tends to one of the stationary points
of type (iv) (lying in ${\mathcal L} < 0$) in Lemma \ref{zeros}.

\end{rmk}

We have therefore deduced the following theorem.
\begin{thm} \label{asymptotics}
The metric corresponding to our trajectory has the form, to leading order in $t$
as $t \rightarrow +\infty$,
\[
dt^2 + {\alpha}^2 d \theta^2 + t \; {\sf h}_{\infty}
\]
where $\alpha$ is a positive constant, $\theta$ is the angle coordinate
on $M_1 = S^1$ and ${\sf h}_{\infty}$ is the product
Einstein metric on $M_2 \times \cdots \times M_r$.
The volume growth is asymptotically $t^{\frac{n+1}{2}}$. \qed
\end{thm}

\begin{rmk}
We thus obtain asymptotic behaviour which is a mixture of the
asymptotically paraboloid geometry of the Bryant solitons on $\mathbb R^n$
(for $n>2$) and the Hamilton-Witten cigar geometry on $\mathbb R^2$.
\end{rmk}

\begin{thm} \label{mainthm}
Let $M_2, \ldots, M_r$ be compact Einstein manifolds with positive scalar
curvature. There is an $r-1$ parameter family of non-homothetic complete smooth
steady Ricci solitons on the trivial  rank $2$ vector bundle
over $M_2 \times \ldots \times M_r$, with asymptotics given by
Theorem \ref{asymptotics}.  \qed
\end{thm}

\begin{rmk} \label{ricci}
As with the metrics of \cite{DW2}, we can show that our soliton metrics
have nonnegative Ricci curvature.
The sectional curvatures decay like $\frac{1}{t}$
or faster, and the curvatures $K(U_i \wedge U_j)$ where $U_i, U_j$
are tangent to $M_i, M_j$ respectively with $i,j \geq 2$ and $i \neq j$
are negative. The scalar curvature decays like $\frac{1}{t},$
and  satisfies $\frac{c_1}{t} \leq \bar{R} \leq \frac{c_2}{t}$ for
certain positive constants $c_1, c_2$ and all sufficiently large $t$.
(These constants depend only on $n$ and $\sqrt{-C}$.)
In particular, the asymptotic scalar curvature ratio
$\limsup_{d \rightarrow +\infty} {\bar{R}} d^2$,  where $\bar{R}$ is the
scalar curvature and $d$ is the distance from a fixed origin
in the manifold, is $+\infty$, as it should be.
\end{rmk}

\section{\bf Complete Ricci-flat metrics}

As mentioned in the introduction, a special case of solutions to the soliton
equations is that of trivial solitons, where the metric is Einstein
and the potential is constant. In the steady case, this means the
metric is Ricci-flat.

In \cite{Bo1} B\"ohm constructed an $r-2$ parameter family of complete
Ricci-flat metrics using warped products over $r$ Einstein manifolds
with positive Einstein constants. He assumed in his construction that the collapsing
Einstein factor is a sphere of dimension at least $2$. In this section we
will remove this dimension restriction, i.e., we produce
analogues of these metrics in the case where $M_1 = S^1$ (so $d_1 =1$).
The special case of $r=2$ was treated in \cite{BB} (see also p. 271 of \cite{Be}),
where an explicit solution was found. It includes the Riemannian Schwarzschild
solution, which is the special case when $M_2 = S^2$.

We recall from \S 2 that for trajectories representing Ricci-flat metrics, we
have ${\mathcal L}=0$ and ${\mathcal H}=1$. Therefore we need to study
trajectories emanating from the critical point $P_0$ and lying in the locus
${\mathcal L} = 0$ rather than going into the region ${\mathcal L} < 0$.
These form an $r-2$ parameter family.

We note that as ${\mathcal L} =0$ we have to modify our procedure to recover the
metric from solutions to (\ref{eqnX}) and (\ref{eqnY}). We now define $t$ by
\begin{equation} \label{stE}
dt = \exp \left( \int_{s*}^{s} \sum_{j=1}^{r} X_j^2 \right) \; ds
\end{equation}
for some fixed $s^*$. Also, let
\[
g_i = \frac{\sqrt{d_i}}{Y_i} \,\exp \left( \int_{s*}^{s} \sum_{j=1}^{r} X_j^2 \right)
\]
so
\[
\frac{\dot{g}_i}{g_i}\, \exp \left( \int_{s*}^{s} \sum_{j=1}^{r} X_j^2 \right)
= -\frac{Y_i^\prime}{Y_i} + \sum_{j=1}^{r} X_j^2
=\frac{X_i}{\sqrt{d_i}}
\]
and hence
\[
{\rm tr} L = \sum_{i=1}^{r} \; \frac{d_i \dot{g_i}}{g_i}
= {\mathcal H} \exp \left( -\int_{s*}^{s} \sum_{j=1}^{r} X_j^2 \right).
\]
As ${\mathcal H}=1$ for Einstein trajectories,
it follows that $dt = \frac{ds}{{\rm tr}L}$.

Note that we have
\begin{equation}  \label{dmetric}
\frac{\sqrt{d_i}}{\tr L} \frac{X_i}{Y_i^2} = \dot{g_i} g_i,
\end{equation}
which is consistent with our formula in the soliton case.

\medskip

As in the soliton case, we can restrict to the subsystem
obtained by omitting the equation for $Y_1$, and deduce that the
flow is defined for all $s$ since ${\mathcal L}=0$ is compact
for the subsystem. We have $X_i,Y_i >0$ along our trajectories as before.
As ${\mathcal L}=0$, we in fact  have $0 < X_i < 1$ for all $i$.
Note also that the variety $\{{\mathcal H}=1, {\mathcal L}=0 \}$ is smooth.

\begin{lemma}
The Ricci-flat metrics corresponding to our trajectories are complete.
\end{lemma}

\begin{proof}
We have
\[
dt = \exp \left(\int_{s^*}^{s} {\mathcal G} \right) \; ds
\]
where ${\mathcal G} = \sum_i X_i^2 \geq \frac{1}{n} {\mathcal H}^2$ by Cauchy-Schwartz.
Since ${\mathcal H} = 1$ along Einstein trajectories, it follows that
$t$ tends to $\infty$ as $s$ does, proving completeness.
\end{proof}

In order to examine the long time behaviour of the Ricci-flat trajectories,
we need to use a modified form of the Lyapunov function ${\mathscr F}_0$ for the flow
discussed in Corollary \ref{steady-F}. Writing ${\mathscr F}_0$ in terms of
the variables $X_i, Y_i$ (cf (\ref{def-Xi}) and (\ref{def-Yi})) we get
$$ {\mathscr F}_0 = \left(\sum_{i=1}^r \, X_i^2 + \sum_{i=1}^r \, \lambda_i Y_i^2
    - \frac{{\mathcal H}^2}{n} \right) \prod_{i=1}^r  \,
    \left(\frac{\sqrt{d_i}}{Y_i} \frac{1}{\xi} \right)^{-\frac{2d_i}{n}}.  $$
Taking into account the conditions ${\mathcal L}=0, {\mathcal H}=1$ and the
fact that $X_1$ plays a special role in the subsystem, we consider the
following modified Lyapunov function with domain
${\mathcal D}:=\{ {\mathcal L}=0, {\mathcal H}=1 \} \cap \{Y_i > 0 \,(i>1), \,|X_1 -1| < \sqrt{2} \} $:
\begin{equation} \label{mod-F}
\hat{\mathscr F} := \frac{1 - \frac{1}{n-1} (1 - X_1)^2}{\prod_{i=2}^{r}
       \left(\sqrt{\lambda_i Y_i} \right)^{\frac{2d_i}{n-1}}}
      =   \frac{\sum_{i=1}^{r} X_i^2 + \sum_{i=2}^{r} \lambda_i Y_i^2
       -\frac{1}{n-1} \left( {\mathcal H} - X_1 \right)^2}{ \prod_{i=2}^{r}
        \left(\sqrt{\lambda_i Y_i} \right)^{\frac{2d_i}{n-1}}}.
\end{equation}
Note that $\hat{\mathscr F}$ is positive along our trajectories as $0 < X_1 < 1$.

\begin{lemma} \label{F-decr}
$\hat{\mathscr F}$ is non-increasing along the trajectories of the flow lying in $\mathcal D$.
\end{lemma}
\begin{proof}
After some algebra we find
\[
\frac{1}{2} \frac{\hat{\mathscr F}^{\prime}}{\hat{\mathscr F}}
= \frac{X_1(1-X_1)({\mathcal G}-1) +(n-2 + 2X_1 - X_1^2)(\frac{1-X_1}{n-1} - {\mathcal G})}
{n-2 + 2X_1 - X_1^2}
\]
where ${\mathcal G} = \sum_{i=1}^{r} X_i^2$ as usual.
For our trajectories the denominator is positive. The numerator
may be rewritten as
\[
\frac{1-X_1}{n-1}\, (n-2 - (n-3)X_1 - X_1^2) + {\mathcal G}(-X_1 + 2-n),
\]
in which the term multiplying $\mathcal G$ is negative.
Now, using Cauchy-Schwartz and ${\mathcal H} =1$, we have the inequality
\[
 {\mathcal G}  \geq X_1^2 + \frac{1}{n-1} \left( \sum_{i=2}^{r} \sqrt{d_i} X_i \right)^2
= X_1^2 + \frac{(1-X_1)^2}{n-1}.
\]
Substituting into the above expression for the numerator, we find
after simplification that the numerator is $\leq X_1^2 (2-n - X_1)$
which is $\leq 0$.
\end{proof}

\begin{rmk} \label{Fstat}
We have $\hat{\mathscr F}^{\prime} =0$ iff $X_1=0$ and we have equality in
Cauchy-Schwartz, that is, when $X_i = \frac{\sqrt{d_i}}{n-1}$ for $i \geq 2$.
\end{rmk}

\begin{lemma} \label{Fmin}
The function $\hat{\mathscr F}$ has a unique critical point in $\mathcal D$
which is the global minimum point.
\end{lemma}

\begin{proof}
We use the second expression of $\hat{\mathscr F}$ in (\ref{mod-F}). By Cauchy-Schwartz
and $X_1 \geq 0$  the numerator is at least $\sum_{i=2}^{r} \lambda_i Y_i^2$.
Next, using similar calculations to those in Prop 4.10 of \cite{DHW} we find that
$\hat{\mathscr F} \geq (n-1) \prod_{i=2}^{r} d_i^{-\frac{d_i}{n-1}}$ in $\mathcal D$.
Equality holds exactly at the point $E$ whose coordinates are given by
\[
X_1=0, \; \;  X_i =\frac{\sqrt{d_i}}{n-1}, \,\,
Y_i = \sqrt{\frac{n-2}{\lambda_i}}\, X_i  \; \; : \;\; (i=2, \ldots r) ;
\]
in fact it is easy to check that $E$ is the unique critical point of $\hat{\mathscr F}$
in $\mathcal D$.
\end{proof}

We can now use $\hat{\mathscr F}$ to analyse the long-time behaviour of the flow.
\begin{thm} \label{omega}
The $r-2$ parameter family of Ricci-flat trajectories all converge to $E$ as $s$
tends to infinity.
\end{thm}

\begin{proof}
As usual we work with the subsystem omitting $Y_1$. As $\{{\mathcal L}=0,
{\mathcal H}=1 \}$ is now compact,  for each trajectory $\gamma$
we have an $\omega$-limit set $\Omega$ lying in the level set
$\hat{\mathscr F} = \mu$ where $\mu$ is the infimum of $\hat{\mathscr F}$ along the
trajectory. Notice that $\mu > 0$ by Lemma \ref{Fmin}, and from the expression
of $\hat{\mathscr F}$ none of the $Y_i$ coordinates of a point in $\Omega$ can be zero.
Hence $\Omega$ lies in $\mathcal D$. As $\Omega$ is flow-invariant, Remark \ref{Fstat} shows
that on $\Omega$ we have $X_1 =0$ and $X_i =\frac{\sqrt{d_i}}{n-1}$ for $i \geq 2$.
In particular, $Y_i^{\prime}$ must vanish. Again by flow-invariance,  we also need
$X_i^{\prime}$ to vanish, and this now forces $\Omega$ to be $\{ E \}$,
as all other stationary points in ${\mathcal L} = 0$ do not lie in ${\mathcal D}$.

So $\mu = \hat{\mathscr F}(E)$, the global minimum of $\hat{\mathscr F}$
in $\mathcal D$. Now let $\epsilon > 0$ be sufficiently small so that the
$\epsilon$-ball around $E$ in ${\mathcal L}=0, {\mathcal H}=1$ is contained
in the region where all $Y_i >0$. Recall that ${\mathcal L}=0, {\mathcal H}=1$
is smooth at $E$. From Lemma \ref{Fmin} we know $E$ is the unique point where
$\mu$ is attained. Therefore the minimum of $\hat{\mathscr F}$ on the $\epsilon$-sphere
around $E$ is $\mu + \delta$ for some $\delta>0$. As $E$ is the
$\omega$-limit set, there exists a time $s_*$ where the trajectory
lies in the open $\epsilon$-ball and $\hat{\mathscr F} (\gamma(s_*)) < \mu + (\delta/2)$.
Now by monotonicity of $\hat{\mathscr F}$ the trajectory can never pass back through the
$\epsilon$-sphere, so is trapped for all later time in the $\epsilon$-ball.
Hence the trajectory converges to $E$.
\end{proof}

\begin{rmk}
One can give an alternative proof of B\"ohm's existence result for complete Ricci-flat metrics
on multiply warped products with $d_1 > 1$ along the above lines by using instead the Lyapunov
function
\[
{\sf F} = \prod_{j=1}^{r} \, Y_j^{-\frac{2d_j}{n}}.
\]
$\sf F$ is again positive and non-increasing along the trajectories of the flow on the
locus where ${\mathcal H} =1, {\mathcal L} =0, X_i, Y_i >0 \, (i=1, ..., r)$. In this set, ${\sf F}$ has a
unique critical point, which is a global minimum, whose coordinates are
$X_i = \frac{\sqrt{d_i}}{n},\, Y_i = \sqrt{\frac{n-1}{\lambda_i}} X_i$. This point
corresponds geometrically to the Ricci flat cone on the product Einstein metric
of $S^{d_1} \times M_2 \times \cdots \times M_r$. An account of this alternative proof
can be found in the McMaster M. Sc. thesis of Cong Zhou.
\end{rmk}

We now consider the asymptotics of the complete Ricci-flat metrics we have constructed.
Note that ${\mathcal G} = \sum_{i=1}^{r} X_i^2$ equals $\frac{1}{n-1}$ at $E$. So we can choose
a sufficiently small positive $\delta$ and $s_1 > s^*$ such that $|{\mathcal G} - \frac{1}{n-1}| < \delta$
for all $s \geq s_1$. Equation (\ref{stE}) then gives us estimates
\[
\frac{\rho_0 (n-1)}{1 - \delta(n-1)} \, \left(e^{(\frac{1}{n-1} - \delta)(s-s_1)}  -1 \right) < t-t_1 <
\frac{\rho_0 (n-1)}{1 + \delta(n-1)} \, \left(e^{(\frac{1}{n-1} + \delta)(s-s_1)}  -1 \right)
\]
where $\rho_0$ is the constant $\exp (\int_{s^*}^{s_1} {\mathcal G})$, and $t_1$ corresponds to $s_1$ via (\ref{stE}).

\begin{lemma} \label{asymp-g1}
The function $g_1(t)^2$ is increasing and bounded from above and
hence converges to a positive constant as $t$ tends to infinity.
\end{lemma}

\begin{proof} That $g_1(t)^2$ is increasing follows from the formula (\ref{dmetric}).
Integrating this we obtain
$$ \frac{1}{2} (g_1(t)^2 - g_1(t_1)^2) = \int_{s_1}^{s} \, \frac{X_1}{Y_1^2}
        \, \exp\left(2 \int_{s^*}^{\sigma} \, \sum_j \, X_j^2   \right) d \sigma.$$
We shall estimate the integral by estimating $X_1, 1/Y_1^2$ and the exponential
separately.

The equation for $X_1$ implies that $ X_1^{\prime} \leq X_1 \left( -\frac{n-2}{n-1} + \delta  \right),$
which yields upon integration
$$ X_1(s) \leq X_1(s_1) \exp \left(-\left(\frac{n-2}{n-1}-\delta \right)(s-s_1)  \right). $$
The equation for $Y_1$ gives
$$ (\log Y_1)^{\prime} = \sum_{j=2}^{r} X_j^2 + X_1^2 - X_1 \geq \frac{(1-X_1)^2}{n-1} + X_1^2 - X_1
      = \frac{1}{n-1} - \left(\frac{n+1}{n-1}\right) X_1 + \frac{n}{n-1} X_1^2,$$
where we have used the Cauchy-Schwartz inequality and ${\mathcal H} =1$. Since $X_1$ tends to $0$
as $s$ tends to infinity, we may assume that $s_1$ has been also chosen so that the absolute value
of the terms involving $X_1$ in the above is less than $\delta$. Integration of the inequality then gives
$$ \frac{1}{Y_1(s)^2} < \frac{1}{Y_1(s_1)^2} \exp\left(-2(s-s_1)\left(\frac{1}{n-1} -\delta \right)\right). $$

Finally,
$$ \exp \int_{s^*}^{\sigma} \sum_j \,X_j^2  = \left(\exp \int_{s^*}^{s_1} {\mathcal G} \right)
       \left( \exp \int_{s_1}^{\sigma} {\mathcal G} \right) \,\leq
       \,\rho_0 \, \exp \left((\sigma -s_1)\left(\frac{1}{n-1}+ \delta \right)\right),$$
where $\rho_0 = \exp(\int_{s^*}^{s_1} {\mathcal G})$ (which depends in particular on the choice of $\delta$).

Now combining the three inequalities we get
$$ \frac{1}{2} (g_1(t)^2 - g_1(t_1)^2) < \rho_0^2\, \left( \frac{X_1(s_1)}{Y_1(s_1)^2}\right)
         \int_{s_1}^{s} \exp \left(\left(-\frac{n-2}{n-1} + 5\delta \right)(\sigma - s_1)\right) d\sigma. $$
As $\delta$ can be chosen arbitrarily small, it follows that $g_1(t)^2$ is bounded above for all $t$.
\end{proof}

Similarly, arguing as in the soliton case, and using the fact
that $\lim_{s \rightarrow \infty} \frac{X_i}{Y_i^2} = \frac{X_i}{Y_i^2}(E)
=\frac{n-1}{n-2} \frac{\lambda_i}{\sqrt{d_i}}$ for $i \geq 2$, we obtain
estimates for $g_i^2(t)- g_i^{2}(t_1)$ for all $t > t_1$. These imply that asymptotically
\begin{equation} \label{cone-asymp}
  c_1 t^{2-\epsilon_0} < g_i(t)^2 < c_2 t^{2+\epsilon_0}
\end{equation}
for arbitrarily small $\epsilon_0 > 0$ and positive constants $c_1, c_2$
depending on $\epsilon_0$ and $\delta$.

\begin{rmk} The asymptotics obtained above are an analogue of those of the
Riemannian Schwarzschild metric, which is the case where $r=2$ and $M_2 = S^2$
(with the constant curvature $1$ metric).

We note also that $Y_1$ tends to infinity (exponentially fast in $s$). So in the
full phase space, the trajectories of the Ricci-flat metrics are indeed unbounded.
\end{rmk}

We have therefore proved

\begin{thm} \label{thm-ricci-flat}
Let $M_2, \ldots, M_r$ be closed Einstein manifolds with positive scalar
curvature. There is an $r-2$ parameter family of non-homothetic complete
smooth Ricci flat metrics of form $($\ref{metric}$)$ on the trivial rank $2$
vector bundle over $M_2 \times \ldots \times M_r$.  Asymptotically, $g_1(t)$
tends to a positive constant and $g_i(t)^2, i>1$  are approximately quadratic
in the sense of $($\ref{cone-asymp}$)$.  \qed
\end{thm}

\begin{rmk} \label{superpot}
The cohomogeneity one Einstein equations can be viewed as the flow on the zero energy
hypersurface of a certain Hamiltonian ${\sf H}$ constructed in \cite{DW4}, \S 1. Recall
that a superpotential $\Phi$ of ${\sf H}$ is a $C^2$ function on the full momentum phase
space such that the equation ${\sf H}(q, d\Phi_q) = 0$ holds. It is shown in \cite{DW5} that
a superpotential automatically gives rise to a first order subsystem of the cohomogeneity
one Einstein equations. In the Ricci-flat case, physicists have frequently been able to show
that solutions to the first order subsystem represented metrics with special holonomy
(see, e.g., \cite{CGLP1} and \cite{CGLP2}).

On the other hand, there are examples of superpotentials which are not associated with
special holonomy, but nevertheless are related to some degree of integrability of
the Ricci-flat equations. We mention here case (1) of Theorem 6.3 and Examples 8.2 and 8.3
in \cite{DW4}. The hypersurfaces in these examples are respectively the product
of one, two, or three Einstein manifolds with positive scalar curvature. The dimensions
of the factors in the latter two cases are restricted, i.e., up to permutation, they
must be $(6,3), (8,2), (5,5)$ and $(3, 3, 3), (4, 4, 2), (6, 3, 2)$ respectively.
With an appropriately chosen sphere as one of the Einstein factors, there are {\em explicit}
solutions of the first order subsystem which are complete smooth Ricci-flat metrics,
and these must occur among the B\"ohm metrics since $d_1 > 1$ is satisfied.

It turns out that if we take a product of the above examples with a circle, we obtain
hypersurfaces with superpotentials as well (cf Remark 2.8 of \cite{DW4}, which applies
also to the null case). However, there are two differences. First, the convex polytopes
associated to the scalar curvature function and superpotentials are no longer of maximal
dimension. This explains why these examples did not occur in the classifications in
\cite{DW6} and \cite{DW7}.  Second, since the circle can be taken to be the collapsing factor,
none of the positive Einstein factors need to be spheres anymore in order to obtain
complete smooth solutions of the first order subsystem. These Ricci-flat metrics
must occur among those constructed in this section. Case (1) of Theorem 6.3 now becomes
the $r=2$ case, which is known to be explicitly integrable (cf \cite{BB}).
\end{rmk}

\medskip

The topology of the underlying manifolds where we have constructed steady soliton
and Ricci-flat structures is also very interesting. We shall consider here the $r=2$
case briefly.

For the Einstein factor $M_2$ we can take the Kervaire sphere $\Sigma$
of dimension $q=4m+1$ with $m \neq 0, 1, 3, 7, 15$ and possibly $31$, or one of the
homotopy spheres in dimension $7, 11, 15$ which bound a parallelizable manifold.
The solution of the Arf-Kervaire invariant problem by Hill, Hopkins, and Ravenel \cite{HHR} implies that
in the above dimensions the Kervaire sphere is not diffeomorphic to the standard sphere.
At the same time, the work of Boyer, Galicki, and Kollar \cite{BGK1}, \cite{BGK2} provides
continuous families of Sasakian Einstein metrics on these homotopy spheres.
On the other hand, it is known (Theorem 1 in \cite{KS}) that for a non-standard homotopy $q$-sphere
$\Sigma$ that bounds a parallelizable manifold, $\R^2 \times \Sigma$ is not diffeomorphic to
$\R^2 \times S^q$. Therefore, Theorems \ref{thm-ricci-flat} and \ref{mainthm} imply

\begin{cor} \label{diffstr}
In dimensions $9, 13, 17$ and all dimensions $4m+3$ with $m \neq 0, 1, 3, 7, 15, 31$
there exist pairs of homeomorphic but not diffeomorphic manifolds both of which admit
a complete Ricci-flat metric. The same conclusion holds for non-Einstein, complete,
steady gradient Ricci soliton structures.
\end{cor}

We are not aware of examples of this type in the literature, although they presumably
occur among Calabi-Yau manifolds. The soliton case is somewhat
surprising in view of the comparatively greater rigidity of the soliton equations.

\section{\bf Numerical examples}

In this section we shall produce some numerical solutions of the equations
(\ref{TT})-(\ref{TS}). We begin with the case of triple warped products
for which we obtained an existence proof in \S 3. The purpose here is to
see how quickly the characteristic asymptotics of steady gradient Ricci solitons
given in \S 1 and \S 3 exhibit themselves. We then produce numerical solutions in two
cases for which we do not yet have an analytic existence proof. The principal orbits
in these cases are the twistor space of quaternionic projective space (viewed
as a quaternionic K\"ahler manifold) and the total space of the corresponding
${\rm Sp}(1)$ bundle. For these examples, the numerics seem to indicate strongly
that complete steady soliton structures do exist.

The procedure we use is the same as that described in \cite{DHW}, \S 5. First,
a series solution is generated for these equations in a neighbourhood of
the singular orbit satisfying the appropriate smoothness conditions. We then
truncate the series and use the values of the resulting functions at some small
$t_0 > 0$ as initial values to generate solutions of the equations for $t > t_0$
via a fourth order Runge-Kutta scheme. This procedure is necessary because
the cohomogeneity one gradient  Ricci soliton equation has an irregular singular point
at $t=0$, the position of the singular orbit. Since non-trivial steady solitons are
necessarily non-compact, we rely on checks against the expected asymptotic behaviour
(cf \S 1) to ascertain that the numerical solutions indeed point to an actual solution.

\medskip

\noindent{\bf A. Triple Warped Product Case:}

\medskip

Taking $r=3$ for a metric of the form (\ref{metric}), the equations (\ref{TT}), (\ref{SS})
and the conservation law (\ref{cons1}) become the system

\begin{eqnarray*} \label{tripleEQ}
\dot{z_1} & = & z_2  \\
\dot{z_2} & = & -d_2 \frac{z_2 z_4}{z_3} - d_3 \frac{z_2 z_6}{z_5} + z_2 z_8 + \frac{\epsilon}{2}z_1 \\
\dot{z_3} & = & z_4 \\
\dot{z_4} & = & -(d_2-1)\frac{z_4^2}{z_3} - \frac{z_2 z_4}{z_1} -d_3 \frac{z_4 z_6}{z_5} + z_4 z_8
        + \frac{d_2 -1}{z_3} + \frac{\epsilon}{2}z_3 \\
\dot{z_5} & = & z_6  \\
\dot{z_6} & = &  -(d_3-1)\frac{z_6^2}{z_5} - \frac{z_2 z_6}{z_1} -d_2 \frac{z_4 z_6}{z_3} + z_6 z_8
        + \frac{d_3 -1}{z_5} + \frac{\epsilon}{2}z_5 \\
\dot{z_7} & = & z_8 \\
\dot{z_8} & = & -z_8 \left( \frac{z_2}{z_1} + d_2 \frac{z_4}{z_3} + d_3 \frac{z_6}{z_5} \right)
            + z_8^2 + \epsilon z_7 + C
\end{eqnarray*}
where $(z_1, \ldots, z_8) :=(g_1, \dot{g_1}, g_2, \dot{g_2}, g_3, \dot{g_3}, u, \dot{u})$ and we have
chosen $\lambda_2 = d_2 -1, \lambda_3 = d_3 -2$.

For steady solitons, we set $\epsilon = 0$ and we may rule out homothetic solitons by
choosing $C$ to be $-1$. Recall from \cite{DHW}, \S 2 that the smoothness conditions
together with the conservation law (\ref{cons1}) yield the general relation
\begin{equation} \label{initial-u}
  (d_1 + 1)\; \ddot{u}(0) = C + \epsilon u.
\end{equation}
Since $d_1 = 1$ and $\epsilon = 0$, we can further set $u(0)= 0$, and (\ref{initial-u}) becomes
 $\ddot{u}(0) = -\frac{1}{2}$. The initial values of $(z_1, \ldots, z_8)$ are then
 $(0, 1, a, 0, b, 0, 0, 0)$ with $a, b > 0$. (Recall that Theorem \ref{mainthm} gives a
 $2$-parameter family of solutions.) The stepsize in the Runge-Kutta scheme is $0.001$,
 so that the accumulated error is of the order $0.001^4$.

\medskip

\noindent{\bf Example 1.} \: We take $d_2=2$ and $d_3=3$, so the cohomogeneity
one manifold has dimension $7$. We include below two sketches of the solution with
initial values $a=6$ and $b=3$. Figure $1$ shows the functions $g_i$ and $u$ over
the range $0 \leq t \leq 10$, and Figure 2 does it for the range $0 \leq t \leq 500$.
Notice that the slope of the potential $u$ quickly becomes approximately equal
to $-1=-\sqrt{-C}$, cf Prop \ref{steady-MC}. The function $g_1$ also quickly
becomes approximately  constant. The limiting value can be made close to $0$ by
 making $a, b$ small. It also appears that regardless of the values of $a, b$,
 this limiting value is bounded above by $2$.

\begin{figure}[hbtp]
\includegraphics[width=15cm,height=10cm]{PictureSum1Zoomedin}
\caption{}
\end{figure}

\pagebreak

\begin{figure}[hbtp]
\includegraphics[width=15cm,height=8cm]{PictureSum1}
\caption{}
\end{figure}

We also plot the quantities $ {\tilde X}_i := \frac{X_i}{\sqrt{d_i}},
\;\;\;  {\tilde Y}_i:=\frac{Y_i}{\sqrt{d_i}}$ against $t$ respectively
in Figures 3 and 4. Recall that the proof of Theorem \ref{asymptotics}
shows that ${\tilde Y}_1$ tends to a positive constant while the remaining
quantities tend to zero.

\begin{figure}[hbtp]
\includegraphics[width=15cm,height=9cm]{PictureSum3}
\caption{}
\end{figure}

\begin{figure}[hbtp]
\includegraphics[width=15cm,height=10cm]{PictureSum4}
\caption{}
\end{figure}

\pagebreak

\noindent{\bf B. Some Two-summands Cases:}

\medskip

We consider next a principal orbit $G/K$ whose isotropy representation consists of two
inequivalent ${\rm Ad}(K)$-invariant irreducible real summands. We assume that
$K \subset H \subset G$ where $H, K$ are closed subgroups of the compact Lie group $G$
such that $H/K$ is a sphere. We choose a $G$-invariant background metric $\mathsf b$ on $G/K$
such that it induces the constant curvature $1$ metric on $H/K$. Our cohomogeneity one
manifold $M$ is then the vector bundle $G \times_{H} \R^{d_{1}+1}$ where $H/K \subset \R^{d_{1}+1}$
is regarded as the unit sphere.

Let $\g = \kf \oplus \p$ be an ${\rm Ad}(K)$-invariant decomposition of the Lie algebra
of $G$. Then $\p$ is identified with the tangent space of $G/K$ at the base point.
We can further decompose $\p$ as $\p_1 \oplus \p_2$, where $\p_i$ are $\mathsf b$-orthogonal
irreducible ${\rm Ad}(K)$-representations of dimensions $d_i$. Note that $\p_1$ is
the tangent space to the sphere $H/K$ at the base point, and $\p_2$ is the tangent
space of the singular orbit $G/H$ at the base point. Our metrics of cohomogeneity one
then take the form
$$ \bar{g} = dt^2 + g_1(t)^2 \; {\mathsf b}|\p_1 + g_2(t)^2 \; {\mathsf b}|\p_2.$$

As is well-known, the Ricci operator of the metric $g(t)$ on $G/K$ has corresponding
components
$$ r_1 = \frac{A_1}{d_1} \frac{1}{g_1^2} + \frac{A_3}{d_1} \frac{g_1^2}{g_2^4} $$
$$ r_2 = \frac{A_2}{d_2} \frac{1}{g_2^2} - \frac{2A_3}{d_2} \frac{g_1^2}{g_2^4}. $$
As a result of our choice of $\mathsf b$, one has $A_1=d_1(d_1-1)$, and the constants $A_2, A_3$
are analogous (but not always equal) to the quantities $c_Q$ and $\|{\mathcal A}\|$ in \cite{DHW}.

Taking $(z_1, \ldots, z_6):=(g_1, \dot{g_1}, g_2, \dot{g_2}, u, \dot{u})$ as before, the
soliton equations become
\begin{eqnarray*} \label{2summands}
\dot{z_1} & = & z_2  \\
\dot{z_2} & = & -(d_1 -1)\frac{z_2^2}{z_1} -d_2 \frac{z_2 z_4}{z_3} + z_2 z_6 + \frac{d_1 -1}{z_1}
            + \frac{A_3}{d_1} \frac{z_1^3}{z_3^4} + \frac{\epsilon}{2}z_1 \\
\dot{z_3} & = & z_4 \\
\dot{z_4} & = & -d_1 \frac{z_2 z_4}{z_1} -(d_2-1)\frac{z_4^2}{z_3} + z_4 z_6
        + \frac{A_2}{d_2} \frac{1}{z_3} -2\frac{A_3}{d_2} \frac{z_1^2}{z_3^3} + \frac{\epsilon}{2}z_3 \\
\dot{z_5} & = & z_6 \\
\dot{z_6} & = & -z_6 \left( d_1 \frac{z_2}{z_1} + d_2 \frac{z_4}{z_3} \right)
            + z_6^2 + \epsilon z_5 + C,
\end{eqnarray*}
with $\epsilon=0$ and $C=-1$.

\medskip

\noindent{\bf Example 2.} \: We set $G={\rm Sp}(m+1),  H={\rm Sp}(m) \times {\rm Sp}(1),$
and $K={\rm Sp}(m) \times {\rm U}(1)$. The principal orbit $G/K$ is diffeomorphic to $\C{\PP}^{2m+1}$
and the singular orbit $G/H$ is $\HH{\PP}^m$. So $d_1=2, d_2 = 4m$, and $A_2 = 2m(m+2), A_3=\frac{m}{2}$
(where $\mathsf b$ is given by $-2\tr(XY)$). As before we set $u(0)=0$ and (\ref{initial-u}) becomes
$\ddot{u}(0) = -\frac{1}{3}$. The initial values of $(z_1, \ldots, z_6)$ are given by
$(0, 1, \bar{h}, 0, 0, 0)$ where $\bar{h} > 0.$

We plot the functions $g_i$ and $u$ for the $m=1$ and $m=2$ cases respectively
in Figures $5$ and $6$. Recall that the dimensions of the cohomogeneity one
manifolds are respectively $7$ and $11$ for the $m=1$ and $m=2$ cases.
The initial value $\bar{h}$ is taken to be $6$ in both cases. Again the soliton
potential very quickly becomes approximately linear with slope close to $-1$.
The growth of the functions $g_i$ is consistent with being  like
$\sqrt{t}$ asymptotically.

\pagebreak

\begin{figure}[hbtp]
\includegraphics[width=15cm,height=9cm]{PictureSum5}
\caption{}
\end{figure}

\begin{figure}[hbtp]
\includegraphics[width=15cm,height=9cm]{PictureSum6}
\caption{}
\end{figure}

When the isotropy representation of the principal orbit is multiplicity free,
we can still define the new variables $s$ and $X_i, Y_i$ as in \S 2.
We again let ${\tilde X_i}= X_i/\sqrt{d_i}$ and ${\tilde Y_i}= Y_i/\sqrt{d_i}$.
Now we can plot the quantities ${\tilde X}_i, {\tilde Y}_i$ against $t$. Figure $7$
corresponds to the $m=1$ case and Figure $8$ to the $m=2$ case. All
these quantities appear to tend to zero, but the ${\tilde X}_i$ do so faster.
Note that since by Proposition \ref{steady-MC} the quantity $\xi$ tends to $1$,
the fact that the ${\tilde X}_i$ appear to tend to zero very likely indicates
that the shape operator $L$ also tends to $0$.

\begin{figure}[hbtp]
\includegraphics[width=17cm,height=9cm]{PictureSum7}
\caption{${\tilde X_i}, {\tilde Y_i}$ for $m=1$}
\end{figure}

\begin{figure}[hbtp]
\includegraphics[width=17cm,height=9cm]{PictureSum8}
\caption{${\tilde X_i}, {\tilde Y_i}$ for $m=2$}
\end{figure}

We can also plot the ratios ${\tilde X}_1/{\tilde X_2}=\frac{\dot{g_1}}{g_1} \frac{g_2}{\dot{g_2}}$
 and ${\tilde Y}_1/{\tilde Y}_2 = \frac{g_2}{g_1}$ against $t$. For the $m=1$ case this is
shown in Figure 9. The first ratio appears to tend to $1$, which likely indicates that the
principal curvatures of the principal orbits asymptotically become equal. It is not as
convincing (from considering other values of $m$) that the second ratio tends to
$1$, although it does appear to tend to some limit.

\begin{figure}[hbtp]
\begin{center}
\includegraphics[width=16cm,height=8cm]{PictureSum9}
\caption{${\tilde X_1}/{\tilde X_2}$ and ${\tilde Y_1}/{\tilde Y_2}$ for $m=1$}
\end{center}
\end{figure}

Similar numerical results hold for larger values of $m$.

\medskip

\noindent{\bf Example 3.} \: We set  $G={\rm Sp}(m+1) \times {\rm Sp}(1),
H={\rm Sp}(m) \times {\rm Sp}(1) \times {\rm Sp}(1),$ and $K={\rm Sp}(m) \times \Delta{\rm Sp}(1)$.
The principal orbit $G/K$ is diffeomorphic to $S^{4m+3}$ and the singular orbit $G/H$ is
again $\HH{\PP}^m$. So $d_1=3, d_2 = 4m$, and $A_2 = 4m(m+2), A_3=\frac{3m}{4}$
(where $\mathsf b$ is given by $-\tr(XY)$ on both the ${\rm Sp}(m+1)$ and ${\rm Sp}(1)$
factors). As before we set $u(0)=0$ and (\ref{initial-u}) becomes
$\ddot{u}(0) = -\frac{1}{4}$. The initial values of $(z_1, \ldots, z_6)$ are given by
$(0, 1, \bar{h}, 0, 0, 0)$ where $\bar{h} > 0.$

For this case we obtain graphs very similar to those in Example 2.

\medskip

Based on the last two examples, we would conjecture that on the
vector bundles $G \times_H \R^{d_1+1},$ where $(G, H,  K)$ are as above, there is a
$1$-parameter family of non-homothetic complete steady gradient Ricci solitons.
We hope to pursue this and related questions in subsequent work.

redpajama_set_name: RedPajamaArXiv


text:
Curb Your Enthusiasm - Season 9 - Promos, Promotional Photos, Featurette, Guest Cast + Key Art *Updated*
Posted by Nirat Anop at September 29, 2017 0 Comments
Episode 9.01 - Foisted! - Promotional Photos
Episode #81 (season 9, episode 1): "Foisted!"
Debut date: SUNDAY, OCT. 1 (10:00-10:30 p.m. ET/PT)
Larry tries to rid himself of an inept assistant, offends Jeff's barber and gets into hot water over a new project.
Story by Larry David & Jeff Schaffer; directed by Jeff Schaffer.
Larry David Takes a Peek Into Season 9
Episodes to be Superized
Season 9 of Larry David's curmudgeon comedy Curb Your Enthusiasm will feature episodes that run notably longer than the normal half-hour length. Apparently, when your show is off the air for six years, you return with plenty to say. "In the intervening years, I was able to accumulate a lot of ideas, so the shows are kind of… dense," David tells EW. "They're stuffed."
"Almost every show is longer than a half an hour, and some are going to go way, way longer," executive producer Jeff Schaffer tells EW. "When we were writing the outlines, we just kept putting stories and scenes we liked. And then we found lots more fun stuff as we were shooting, so we ended up with way more than 10 episodes worth."
Read More on EW
New Featurette - Back on Set for Season 9 with Larry David, J.B. Smoove & More
Lauren Graham Revealed to Be Playing Larry's Love Interest
Lauren Graham's first gig after reprising her role as Lorelai on the Gilmore Girls revival will be a multi-episode guest arc on Curb Your Enthusiasm. And as it turns out, Larry David will think she's prettay, prettay.
Graham plays an NBC censor who becomes a love interest for the Seinfeld mastermind in the HBO comedy's ninth season. "Larry's always had a complicated relationship with NBC censors. Now he's dating one," Curb executive producer Jeff Schaffer tells EW. "This is a pretty good relationship in spite of the fact that she's an NBC censor. She does have a kid. And that can complicate things."
New Promo - Larry David is the Hero We Need + New Poster
First Look Photos
HIT HBO COMEDY SERIES CURB YOUR ENTHUSIASM, STARRING LARRY DAVID, RETURNS FOR ITS NINTH SEASON OCT. 1
The Emmy®- and Golden Globe-winning comedy series CURB YOUR ENTHUSIASM, starring Larry David, returns for its ten-episode ninth season SUNDAY, OCT. 1 (10:00-10:30 p.m. ET/PT), on HBO. The show stars "Seinfeld" co-creator David as an over-the-top version of himself in an unsparing but tongue-in-cheek depiction of his life.
The new season brings back cast favorites Cheryl Hines as Cheryl, Jeff Garlin as Jeff, Susie Essman as Susie and J.B. Smoove as Leon, as well as series veterans Richard Lewis, Bob Einstein, Ted Danson and Mary Steenburgen.
Also appearing on CURB YOUR ENTHUSIASM this season are Elizabeth Banks, Ed Begley, Jr., Carrie Brownstein, Bryan Cranston, Lauren Graham, Jimmy Kimmel, Nick Offerman, Nasim Pedrad and Elizabeth Perkins.
Season nine directors include Jeff Schaffer, Larry Charles, Robert B. Weide, Jessie Nelson and Bryan Gordon.
CURB YOUR ENTHUSIASM was created by Larry David; executive produced by Larry David, Jeff Garlin and Jeff Schaffer.
Key Art
The Emmy®-winning comedy series CURB YOUR ENTHUSIASM, starring Larry David, returns for its ten-episode ninth season SUNDAY, OCT. 1 (10:00–10:30 p.m. ET/PT), on HBO. The show brings back cast favorites Jeff Garlin, Susie Essman and JB Smoove, among many others. Larry David created and executive produces the series with Jeff Garlin and Jeff Schaffer.

redpajama_set_name: RedPajamaCommonCrawl


text:
EASD. Acarbose Offers No CVD Protection in ACE, but May Cut Diabetes Risk. The Lancet Diab and Endocr
LISBON, PORTUGAL — The α-glucosidase inhibitor acarbose (Glucobay/Precose/Prandase, Bayer) reduces the incidence of diabetes but not of major cardiovascular events among Chinese patients with established heart disease and impaired glucose tolerance (IGT), new research indicates.
Findings from the Acarbose Cardiovascular Evaluation (ACE) trial were presented September 13 here at the European Association for the Study of Diabetes (EASD) 2017 Annual Meeting by several members of the research team, led by Rury R Holman, director of the University of Oxford Diabetes Trials Unit, United Kingdom. The results were simultaneously published in Lancet Diabetes & Endocrinology.
Acarbose, the first α-glucosidase inhibitor approved for the treatment of type 2 diabetes in the 1990s, acts to impede the absorption of carbohydrates and therefore works primarily to reduce postprandial rises in glucose.
It is widely prescribed to treat type 2 diabetes in China and some other Asian countries, but far less so elsewhere.
In the randomized, double-blind, placebo-controlled phase 4 ACE trial of 6522 Chinese adults with IGT and cardiovascular disease, there was no difference in the incidence of a composite 5-point major adverse cardiovascular event (MACE) score between those who took 50 mg of acarbose three times daily or placebo for a median of 5 years. However, the drug did reduce the incidence of new-onset diabetes by 18%, with a number needed to treat of 41 for 5 years.
"These results extend the knowledge of the safety of acarbose and its efficacy for delaying the onset of diabetes to a population with both coronary heart disease and impaired glucose tolerance," Dr Holman said, adding that although the CVD event rate wasn't reduced in the study, "the reduced incidence of diabetes seen with acarbose in the ACE trial might help to reduce cardiovascular risk in the longer term by delaying the onset of diabetes in the high-risk population studied."
But audience member Dr Peter H Bennett, former chief of the Phoenix Epidemiology and Clinical Research Branch of the US National Institute of Diabetes and Digestive and Kidney Diseases, wasn't impressed, even with the diabetes-prevention result.
"The reduction in the incidence of diabetes was only about 18% after 5 years, which I would say is very modest, compared with what you see with just lifestyle interventions," he told Medscape Medical News, noting that the number needed to treat with lifestyle intervention to prevent one case of diabetes in 5 years is only about six or seven.
Writing in an editorial accompanying the published ACE study, Drs Michael A Nauck and Juris J Meier (Diabetes Center Bochum-Hattingen, Ruhr-University Bochum, Germany) agree. "Other preventive measures such as intensive lifestyle intervention, metformin, or thiazolidinediones seem to be more effective in quantitative terms."
And regarding the CVD findings, Dr Bennett said, "Clearly, acarbose is not a good medication for secondary prevention of cardiovascular disease in patients with IGT."
"It [acarbose] might work in people with diabetes, but that's not what this trial addressed," he continued. "It's strictly a secondary-prevention trial of CVD in people with previous heart disease who also happened to have IGT, so it's a restricted population."
However, another audience member, Guangwei Li, MD, chief of the department of endocrinology and cardiology at Fu Wai Hospital and Peking Union Medical College, Beijing, China, had a different take: "I think it's a very important trial, in that it shows no harm for cardiovascular disease. Also, there was no hypoglycemia or increased body weight."
Dr Li told Medscape Medical News that he prescribes acarbose for some of his type 2 diabetes patients, particularly those who are newly diagnosed with only mild glycemic elevation and for whom postprandial blood glucose levels are their primary problem: "It's very useful for them."
He also noted that even though acarbose isn't cheap, costing about the equivalent of a $1 a day, it is extremely popular in China for treating diabetes. It is also indicated for prediabetes there, but not widely used in that population because it's covered by the government reimbursement system only for the treatment of diabetes.
ACE vs STOP-NIDDM
The rationale for the ACE trial had come from the 2002 Study To Prevent Non-Insulin-Dependent Diabetes Mellitus (STOP-NIDDM), which showed that acarbose reduced the incidence of type 2 diabetes by 25% in people with IGT.
A subsequent prespecified secondary analysis suggested a decrease in a composite cardiovascular outcome, although the numbers were low.
At the EASD session, commentator Chantal Mathieu, MD, professor of medicine at the Katholieke Universiteit and chair of endocrinology at the University Hospital Gasthuisberg, Leuven, Belgium, pointed out several differences between STOP-NIDDM and ACE, including in the components of the composite outcome, the populations studied, and the numbers studied.
And Drs Nauck and Meier observe in their editorial: "The results from the ACE trial challenge the conclusions derived from the cardiovascular outcomes reported in STOP-NIDDM, mainly because STOP-NIDDM was not powered for such a secondary analysis and because the results were of borderline significance."
Dr Mathieu also cautioned against comparing the ACE cardiovascular results in people with prediabetes with any of the recent cardiovascular-outcomes trials for various other type 2 diabetes drugs, since those are mostly in people who already have diabetes.
No Difference in Primary Outcome, but an 18% Diabetes Reduction
The ACE composite primary outcome was first occurrence of cardiovascular death, nonfatal myocardial infarction, nonfatal stroke, hospital admission for unstable angina, or hospital admission for heart failure.
The originally planned primary outcome had only included the first three of those five components, but the latter two were added due to slow enrollment and low event rates. The original three-point MACE then became a secondary outcome.
The primary outcome occurred in 470 (14.4%) of 3272 participants in the acarbose group (3.33 per 100 person-years) and 479 (14.7%) of 3250 in the placebo group (3.41 per 100 person-years), with a nonsignificant hazard ratio of 0.98 (P = .73). Hazard ratios didn't differ between the acarbose and placebo groups for any of the five MACE components, nor for the three-component MACE.
On the other hand, diabetes developed less frequently in the acarbose group, occurring in 436 (13%) of 3272 (3.17 per 100 person-years) vs 513 (16%) of 3250 placebo subjects (3.84/100 person-years), with rate ratio 0.82 (P = .005).
At 1 year into the study, the acarbose group also had significantly lower triglycerides (1.49 vs 1.62 mmol/L, P < .0001) and bodyweight (69.9 vs 70.8 kg, P < .0001).
The number of participants reporting mild and severe hypoglycemia didn't differ between the acarbose and placebo groups, 11% and 2%, respectively, for both groups.
Gastrointestinal disorders, including flatulence, were the only adverse event category that differed between the two groups. Serious GI events didn't differ (2% in each group), but GI events leading to discontinuation or dose reduction were more common with acarbose than with placebo (215 [7%] of 3263 vs 150 [5%] of 3241, P = .0007). Death rates did not differ between the groups.
The ACE trial was sponsored by Bayer AG.
Lancet Diabetes Endocrinol. Published online September 13, 2017.
From www.medpage.com


text:
Home » Articole » EN » Games » Gambling » Gambling World » Assessing the proportionality of specific corporate requirements imposed on gambling operators in EU
Assessing the proportionality of specific corporate requirements imposed on gambling operators in EU
by Nicolae Sfetcu | posted in: Gambling World, On-line gambling in EU	| 0
Seat requirements
The CJEU has held that "the requirement that an undertaking create a permanent establishment or branch in the Member State in which the services are provided runs directly counter to the freedom to provide services since it renders impossible the provision of services, in that Member State, by undertakings established in other Member States". (C-546/07 Commission v Germany, par. 39.) The CJEU has furthermore underlined that "if the requirement of authorization constitutes a restriction on the freedom to provide services, the requirement of a permanent establishment is the very negation of that freedom. For such a requirement to be accepted, it must be shown that it constitutes a condition which is indispensable for attaining the objective pursued." (See, inter alia, Case C-452/04 Fidium Finanz AG, ECR [2006] I-09521, par. 46.)
The Court has stated that a requirement for a licence holder to have its seat in the national territory is a discriminatory restriction which can therefore be justified only on the grounds of public policy, public security or public health. (Dickinger and Ömer, p.79, Engelmann, p. 34) The Court has, in this context, underlined that "it is settled case-law that the concept of public policy, first, presumes that there is a genuine and sufficiently serious threat to a fundamental interest of society and, second, must, as a justification for derogation from a fundamental principle of the Treaty, be narrowly construed." (Par. 82 of the Dickinger and Ömer.)
In Engelmann, the national (Austrian) legislation provided that one of the conditions for granting concessions for the operation of land-based games of chance was that operators should have their seat in Austria. The Austrian Government claimed that the purpose of this was to permit effective control of operators in the gambling sector, with a view to preventing those activities from being carried out for criminal or fraudulent purposes. The obligation would permit, in particular, a degree of control to be exercised over the decisions taken by the company's organs by reason of the presence of representatives of the State in organs such as the supervisory board.
The CJEU ruled however that, without it being necessary to determine whether that objective can fall within the definition of public policy, "the categorical exclusion of operators whose seat is in another Member State appears disproportionate, as it goes beyond what is necessary to combat crime. There are indeed various measures available to monitor the activities and accounts of such operators". (Engelmann, par. 37, with reference to, inter alia, the judgments in the cases Gambelli and Placanica.)
The Court referred, inter alia, to the possibility of requiring separate accounts audited by an external accountant to be kept for each gaming establishment of the same operator, the possibility of being systematically informed of the decisions adopted by the organs of the concession holders and the possibility of gathering information concerning their managers and principal shareholders. Any gaming establishment in the Member State can be supervised and have sanctions imposed on it, regardless of the residence of its managers. Member States can also carry out supervision on premises of the gaming establishments located in national territory in order, in particular, to prevent any fraud being committed by operators against consumers. (Engelmann, par. 38-39.)
In Dickinger and Ömer, the Court assessed a similar condition for the exploitation of online casinos. The national (again Austrian) law provided that the holder of the monopoly of operating lotteries must have its registered office in national territory. According to the Austrian Government, this obligation should allow effective monitoring the holder of the monopoly for online casinos, and it is necessary to ensure effective supervision of the operator, inter alia by the presence of State commissioners in its supervisory bodies. The CJEU ruled that it is for the national court to determine, first, whether the objectives in question are capable of falling within the concept of public policy, and, if so, whether the obligation concerning the registered office satisfies the criteria of necessity and proportionality. In particular, it must be considered whether there are other less restrictive means of ensuring a level of supervision of the activities of operators established in another Member State equivalent to that which can be carried out in respect of operators whose registered office is in the country of the provision of the service. (See par. 83 and 84 of Dickinger and Ömer.)
Legal form for operators of games of chance
The CJEU has clarified that the requirement of a particular legal form for operators of games of chance such as a public limited company may, by virtue of the obligations binding certain kinds of company with respect in particular to their internal organisation, the keeping of their accounts, the scrutiny of which they may be subject and their relations with third parties, be justified by the objective of preventing money laundering and fraud. (Dickinger and Ömer, par. 76; Engelmann, par. 28-30.)
Minimum share capital requirements
Similarly, the requirement for a share capital of a certain amount may prove to be of use in order to ensure a certain financial capacity on the part of the operator and to guarantee that it is in a position to meet the obligations contracted towards winning gamblers. The observance of the principle of proportionality requires however that the restriction imposed does not go beyond what is necessary for achieving the aim pursued, having regard to other possible ways of ensuring that the claims of the winning gamblers will be honoured by the operator.(Dickinger and Ömer, par. 77.)
© European Union
corporate requirements, EU, gambling operators
Economics of Gambling in Greece – Lottery
Greece has casinos, two national lotteries, and several types of gambling on football results as...
Gambling in Estonia
Gambling and Lottery issues are regulated on the federal level. The laws listed below are...
The Remote Gambling Industry in the EU
As part of the Study leading up to this Report, a survey instrument was developed...
Bally's Las Vegas
Bally's 3645 Las Vegas Blvd South Las Vegas, NV 89109 Number of rooms 2,814 Theme...
Gambling in Slovenia
Slovenia is law-governed state with civil law system. The territory of the Republic Slovenia is...
Report on the European Las Vegas in Spain (Los Monegros, Aragon): Gran Scala project
Gran Scala is a €17 billion entertainment city to be built on 2,025 hectares in...
Gambling games in Finland
Lotteries Finland's National Lottery is operated by the state owned company Oy Veikkaus, and its...
EU Clear English Tips for Translators – Leave out unnecessary words
Change > to Article 1 of the present Regulation / hereof > Article 1 (of...
▼Games
▼Gambling
▼European Union
The Mystery of the Golden Stars (Le mystère des étoiles d'or) $3.99
Computing Devices $0.00
How to Translate - English Translation Guide in European Union $5.99 – $11.97
Easter Celebration $2.99
Anthologie des meilleures petits contes françaises pour enfants (Anthology of the Best French Short Stories for Children) $4.99
Beauty, Elegance, Creativity - 12 Interviews on the Act of Creation
Problems with translation $0.00


Nutrition is the key to curing more than your need to eat.
Maybe you've heard the locker room mantra: Train hard or go home. Anyone who has ever dedicated themself to a serious workout plan knows that this is true. There's nothing gained from a half-hearted plan. You might as well stay home and binge on YouTube videos.
And yet, being greater than better is more than grunt and sweat! A lot more.
Dr. Philip Goglia has been a certified nutritionist for more than 30 years and is recognized as an elite performance nutrition and rejuvenative health practitioner. He also works with me to help bring the 4D Health concept to my patients – working on the wellness of patients as a first course of action well before we introduce surgical solutions for neck and back problems.
One of the first things that he tells his patients is that fitness and physique is 80% kitchen and 20% gym.
Of course, he's using the word "kitchen" figuratively, but the point is very clear: what you do with nutrition – your intake of food – is the trailhead on the journey to health and wellness. When I hear him explain this to a patient, I think of what the Taoists say about a journey of a thousand miles – it begins with the first step.
1. Avoid foods that have multi-ingredients, such as breads, muffins, bagels, pastries, cereal, and so on. Basically, you want to curtail any baked goods that include yeast, sugar, mold, and gluten.
2. Avoid packaged diet foods that advertise low or no fat and low or no sugar.
3. Hydrate your body. That's water; and you ought to be drinking between one-half ounce to one full ounce of water per each pound you weigh, per day. Do your conversions. If you weigh 150 pounds, you should be drinking between 10 to 19 of water cups! Per day!
4. Although metabolic body type varies from one person to another, across the board, a healthy meal includes high-fat fish, one starchy carbohydrate (potatoes, rice, yams), organic poultry, fresh fruits and fresh vegetables.
5. Try to eliminate as much dairy from your diet as possible. A splash of milk in your coffee is okay, though; one cappuccino per day is fine.
6. Never rely on supplementation over food and exercise. Supplements will not fix a pizza or a skipped meal.
7. Save the biggest protein meal you eat at night several hours before bed; give your body time to tank up on rest and resources to repair your muscle tissue.
8. Need a pre-work out snack? Eat raw nuts – like a handful of almonds with a piece of fruit. About 12 almonds is an excellent source of fat (as an energy source) and fresh fruit is a better source for sugar. The fat/sugar combo will be extremely energizing and allow most people to sustain a 60-90 minute workout.
9. Avoid sports drinks. Many of them are high in sodium and sugar. Many contain caffeine, which will cause digestive discomfort in most people. The type of energy you get from sports drinks will be "spikey" at best and not sustainable (see #8 for a better solution).
10. Dr. Goglia tells me that one of the questions he's often asked is what type of protein bars are best to eat. His answer: "There is no such thing as a protein bar! Does your protein bar taste like salmon, chicken or steak?" Thanks to marketing, the short answer is, "No." The long answer: all of the popular "protein" bars taste like chocolate or cookies; basically, sugar with a fairy dusting of some low-grade protein. If your food is not fish, poultry, steak, eggs or nuts, it's not protein. Period. Dr. Goglia suggests, if you really need a pre-work out bar, look for a healthy split between sugar and fat for sustainable workout energy. He recommends Bonk Breaker bars as his 'go-to' choice that has a good 50/50 split between sugar and fat. I know that many serious cyclists and multisport athletes also carry them around.
Dr. Goglia's last word on workouts and nutrition: he can't emphasize enough on the importance of drinking plenty of water. Eat balanced meals; please do not under eat. Whatever you do, be honest with yourself: don't try to convince yourself that you can turn apple pie into an apple.


Your mind is a very powerful tool.
For an exercise that requires physical strength, I relied more on mental fortitude. Before I took my decision to get strapped up in the safety gear, I had made up my mind that I would complete the course with little hesitance.
Your mind decides primarily what you can and cannot do. It is important to guard it with your life. Be careful of what you allow into your mind. What are you reading, watching, who do interact with? Do you allow another person's opinion of you to become your reality? Most times, to have a strong mind is better than to have strong body. A strong mind controls your whole being. Having both a strong mind and a strong body is priceless grace!
It is okay to not start with the pack or not start when expected to if this helps in preparation.
I must have made my decision to start about 30 mins after everyone else started. I stood at the bottom of the beams and took pictures as each one came up to start the course. Standing there gave me a tangible insight into what it would take to complete the task.
If not starting when you are expected to allows you to plan and prepare your mind, do it. Life is no race. I was more prepared than most once I began and even finished before some. I made up my mind to embrace the challenge and complete gracefully. I did not shout or scream because I told myself I will and had prepared my mind for what lay ahead.
Those who have gone ahead of you, have something you don't yet have: experience.
By the time I was about half-way done with the task at hand, some had long finished and were back on the ground. They were motivating and cheering on those of us yet to complete. After each portion of the course, I shouted down "What is the technique for this stage?" and there was always someone who responded.
Doing the course required a lot of technique. How do you walk on a rope? Angle your foot to the left and right slightly such that the middle portion of your foot is touching the rope. This helps you balance.
In life, there is always someone who has completed that task you are about to undertake. Ask for help. Most people will willing show you how they did it. Learn from the experiences of others.
Tailor techniques to suit yourself: Do what works for you.
While it's true that others who have gone ahead of you have experience, do what works for you. At one point, I realized someone else's technique will not work best for me. I listened to the strategy behind it and adapted it to myself to complete that part.
It is important to recognize the fact that no two people are the same. What worked for the person whose advise you are seeking might not necessarily work for you. However, if you pay close attention to the reason why something was done in a particular way and not always how it was done, you will figure out what works for you.
A colleague shouted this up at me while I was mid-way on a particular portion of the Legon Botanical Gardens Rope Course. This first thing that came to mind was to balance my body before moving. I started to adjust my posture. But then, I heard "Steady yourself internally" and once I did, I took the next step with more confidence. It changed a lot when I was internally steady. I felt the difference. The task at hand became easier. Who you are is more than just your body. Acknowledge this.
Focus on the task at hand but plan ahead.
For each obstacle on the rope course, I looked at it, asked for help where needed and focused on completing it. At the end of the task, I would look further to see what lay ahead to prepare mentally for it. However, my main concentration was finishing what was before me.
I would have been worn out if I went from one task to another without breaks. But at the end of each task, there was a platform. For me, that platform was there for resting, re-strategizing.
Overthinking brings in a flood of negative emotions like doubt, anxiety, fear. Once you know you are capable, go for it. Going through many cycles of ''what if's'' de-motivates you. What if I fall? What if I get stuck in the middle of the rope course? I would not have started if I replayed these questions like a broken record.
It is very important to have this. What are your strengths,weaknesses? How would you leverage on your strengths? What can you do to compensate for your weaknesses? Take time to know yourself. And re-learn how you evolve. Experiences change us to an extent but the core of who we are remains constant.
Get your hands dirty and motivate others.
Sometimes standing on the side line and cheering people on doesn't motivate them much. Take part and others will follow. I stood on the ground below the beams and shouted encouraging words to a colleague. However, he did not budge. When I joined the task, my colleague moved ahead and finished the task successfully.
Let the voices of the crowd in…sometimes.
Listen to the cheering crowd as you go from one step to another. Allow their voices in occasionally, but tune them out occasionally. Focus.
Sometimes the people around you do not know what you are capable of. When they begin to cheer you for the smallest obstacle overcome, do not let this get to your head. Allow it to encourage you to accomplish your full potential. Also, when you are too engaged with the crowd, you might lose focus on the task at hand.
Those around you may see a bigger picture which you can only see if you remove yourself from the task. Someone told me "put more of your weight on your legs, than you arms". I then realized I had put my entire body weight on my arms. I shifted my weight and that made a lot of difference. My arms are aching now for all the time I put my weight on them without realizing. It could have been worse if I did not listen to counsel.
Annatu Abdulai is an impact driven professional with four years of experience supporting early stage high potential businesses in various sectors in Ghana to improve their business models and access growth capital.


VIA: The Force has topped Harry Potter as "Star Wars: The Force Awakens" took in $57 million in opening Thursday-night preview shows in the U.S., setting its first box office record.
Disney hasn't offered a forecast but its marketing campaign has clearly tapped into an enormous pent-up demand for the return of Luke Skywalker, Chewbacca and Han Solo, plus the new characters and story line. The movie is opening six years after Disney paid $4.06 billion for Lucasfilm.


text:
Miguel Angel Luque Fernandez, PhD
Scientific Collaborator, Department of Epidemiology
Harvard T.H. Chan School of Public Health,
677 Huntington Ave. Boston, MA 02215
Educational Note: Paradoxical collider effect in the analysis of non-communicable disease epidemiological data: a reproducible illustration and web application
Luque-Fernandez MA, Schomaker M, Redondo-Sanchez D, Perez MJS, Vaidya A, Schnitzer ME. Educational Note: Paradoxical collider effect in the analysis of non-communicable disease epidemiological data: a reproducible illustration and web application [Internet]. International Journal of Epidemiology 2019;:dyy275.
LuqueFernandez_Collider_IJE_2018.pdf 0 bytes
Classical epidemiology has focused on the control of confounding, but it is only recently that epidemiologists have started to focus on the bias produced by colliders. A collider for a certain pair of variables (e.g. an outcome Y and an exposure A) is a third variable (C) that is caused by both. In a directed acyclic graph (DAG), a collider is the variable in the middle of an inverted fork (i.e. the variable C in A → C ← Y). Controlling for, or conditioning an analysis on a collider (i.e. through stratification or regression) can introduce a spurious association between its causes. This potentially explains many paradoxical findings in the medical literature, where established risk factors for a particular outcome appear protective. We use an example from non-communicable disease epidemiology to contextualize and explain the effect of conditioning on a collider. We generate a dataset with 1000 observations, and run Monte-Carlo simulations to estimate the effect of 24-h dietary sodium intake on systolic blood pressure, controlling for age, which acts as a confounder, and 24-h urinary protein excretion, which acts as a collider. We illustrate how adding a collider to a regression model introduces bias. Thus, to prevent paradoxical associations, epidemiologists estimating causal effects should be wary of conditioning on colliders. We provide R code in easy-to-read boxes throughout the manuscript, and a GitHub repository [https://github.com/migariane/ColliderApp] for the reader to reproduce our example. We also provide an educational web application allowing real-time interaction to visualize the paradoxical effect of conditioning on a collider [http://watzilei.com/shiny/collider/].
Publisher's Version
Tutorial: Introduction to computational causal inference using reproducible Stata, R and Python code
Maternal Dietary Patterns during Pregnancy and Their Association with Gestational Weight Gain and Nutrient Adequacy
Doxorubicin and subsequent risk of cardiovascular diseases among survivors of diffuse large B-cell lymphoma in Hong Kong
Deprivation gap in colorectal cancer survival attributable to stage at diagnosis: A population-based study in Spain
Socioeconomic Inequalities in Colorectal Cancer Survival in Southern Spain: A Multilevel Population-Based Cohort Study