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\address{$^*$ Department of Mathematics, Rutgers University, Piscataway, NJ 08854}

\address{$^{**}$ Department of Mathematics and Computer Science, Rutgers University, Newark, NJ 07102}
\address{$^\dagger$Department of Mathematics and Computer Science,  Rutgers University, Newark, NJ 07102}

\centerline{\bf   AN ANALYTIC PROOF OF THE STABLE REDUCTION THEOREM }

\bigskip

\centerline{ \small  JIAN SONG$^*$, JACOB STURM$^{**}$, XIAOWEI WANG$^\dagger$   \footnote{{ Research supported in part by National Science Foundation grant  DMS-1711439, DMS-1609335 and  Simons Foundation Mathematics and Physical Sciences-Collaboration Grants, Award Number: 631318.}}}

 \bigskip
 \medskip

{\noindent \small A{\scriptsize BSTRACT}. \footnotesize $~$~~  The stable reduction theorem says that a family of curves of genus $g\geq 2$ over a punctured curve can be uniquely completed (after possible base change) by inserting certain stable curves at the punctures. We give a new proof of this result for curves defined over ${\mathbb C}$, using the {\rm K\"ahler-Einstein\ } metrics on the fibers to obtain the limiting stable curves at the punctures.   }

\bigskip

\section{Introduction}

Let $X_1, X_2, ...$ be a sequence of compact Riemann surfaces of genus $g\geq 2$.
A consequence of the Deligne-Mumford construction of moduli space is the following. There exists $N>0$ and imbeddings $T_i: X_i\hookrightarrow {\mathbb P}^N$
such that after passing to a subsequence, $T_i(X_i)=W_i\subseteq{\mathbb P}^N$ converges to a stable algebraic
curve, i.e. a curve $W_\infty\subseteq{\mathbb P}^N$ whose singular locus is either empty or consists of nodes, and whose smooth locus carries a metric of constant negative curvature. The stable reduction theorem \cite{DM} (stated below) is the analogue of this result with
$\{X_i\,:\, i \in \mathbb N\}$ replaced by an algebraic family $\{X_t\,:\, t\in \Delta^*\}$ where $\Delta^*\subseteq{\mathbb C}$ is the punctured unit disk.

\vskip .1in

The imbeddings $T_i$ are determined by a canonical (up to a uniformly bounded automorphism) basis of $H^0(X_i, mK_{X_i})$ (here $m\geq 3$ is fixed). We are naturally led to ask: Can one construct the canonical basis defining $T_i$ explicitly? In Theorem \ref{main} we give an affirmative answer to this question.
\vskip .1in
The main goal of this paper is to give an independent analytic proof of these algebraic compactness results, which is the content of Theorem \ref{1point2}. We start with the Bers compactness theorem, which says that after passing to a subsequence, the $X_i$ converge to a nodal curve in the Cheeger-Colding topology. We then use the technique of Donaldson-Sun \cite{DS} which uses the K\"ahler-Einstein metric to build a bridge between analytic convergence (in Teichmuller space) to algebraic convergence (in projective space). The main difficulty is that unlike the \cite{DS} setting, the diameters of the $X_i$ are unbounded and as a consequence, some of the pluri-canonical sections on $X_\infty$ are not members of $L^2(X_\infty, \omega_{\rm KE})$, so one can't apply the $L^2$-Bergman imbedding/peak section method directly. In order
to solve this problem, we introduce the ``$\epsilon$-Bergman inner product"
on the vector space $H^0(X_i, mK_{X_i})$, which is defined by the $L^2$ norm on the thick part of the $X_i$ (unlike the standard Bergman inner product which is the $L^2$ norm defined by integration on all of $X_i$)
and we show that for fixed $m\geq 3$  the canonical basis defining $T_i$ is an an orthnormal basis for this new inner product. This establishes Theorem \ref{main} which we then use to prove Theorem
\ref{1point2} (the stable reduction theorem).
\vskip .1in
We start by reviewing the corresponding compactness results for Fano maniolds established by Donaldson-Sun in \cite{DS}.\,
Let $(X_i,\omega_i)$ be a  sequence of  K\"ahler-Einstein  manifolds of dimension $n$ with
$c_1>0$, volume at least $V$ and diameter at most $D$, normalized so
that ${\rm Ric}(\omega_i)=\omega_i$.   The first step in the proof of the Donaldson-Sun theorem is the application of Gromov's compactness theorem which implies that after passing to a subsequence,  $X_i$ converges to a compact metric space $X_\infty$ of dimension $n$
in the metric sense, i.e. the Cheeger-Colding (CC) sense. This first step is not not available in the $c_1<0$ case due to the possibility of collapsing and unbounded diameter. Nevertheless, the analogue of this Cheeger-Colding property for Riemann surfaces of genus $g\geq 2$ is available thanks to the compactness theorem of Bers \cite{B}.

\vskip .1in
For the second step, Donaldson-Sun construct explicit imbeddings $T_i: X_i\hookrightarrow{\mathbb P}^N$ with the following properties.
 Let $X_i\rightarrow X_\infty$
 in the Cheeger-Colding sense as above.
Then  there is a K-stable algebraic variety $W_\infty\subseteq{\mathbb P}^N$
such that if $W_i=T_i(X_i)$ then
$W_i\rightarrow W_\infty$ in the algebraic sense (i.e. as points in the Hilbert scheme). Moreover,
$ T_\infty: X_\infty\rightarrow W_\infty
$
is a homeomorphism, biholomorphic on the smooth loci, where

\beta\label{Tinf}\hbox{
$T_\infty(x_\infty)=\lim_{i\to\infty} T_i(x_i)$\ \  whenever\ \  $x_i\rightarrow x_\infty$.
}
\ee
\vskip .1in
We summarize this result with the following diagram:

\beta\label{diag}
\begin{tikzcd}
  & X_i\ \arrow{r}{T_i}\arrow{d}[swap]{\rm CC}
  &\  W_i\ \arrow[hookrightarrow]{r}{}\arrow{d}{\rm Hilb}
  & \ {\mathbb P}^N
  & {} \\
  & X_\infty\ \arrow{r}{T_\infty}
  &\ W_\infty\ \arrow[hookrightarrow]{r}[swap]{}
  & \ {\mathbb P}^N
  \end{tikzcd}
\ee

Here the vertical arrows represent convergence in the metric (Cheeger-Colding) sense and the the algebraic (Hilbert scheme) sense respectively. The  horizontal arrows isomorphisms:  $T_i$ is an algebraic isomorphism, and $T_\infty$ is a holomorphic isomorphism. For $1\leq i\leq \infty$, the maps $W_i\hookrightarrow{\mathbb P}^N$ are inclusions.
\vskip .1in
The imbeddings
$T_i: X_i\rightarrow{\mathbb P}^N$ are the so called
``Bergman imbeddings". This means  $T_i=(s_0,...,s_N)$ where the $s_\alpha$ form an
orthonormal basis of $H^0(X_i,-mK_{X_i})$ with respect to the Bergman inner product:

\beta\label{berg}
\int_{X_i} (s_\alpha,s_\beta)\,\omega_i^n\ = \ \delta_{\alpha,\beta}
\ee

Here  $m$ is a fixed integer which is independent of $i$ and the pointwise inner product is defined by $(s_\alpha,s_\beta)=s_\alpha\bar s_\beta\omega_i^m$.  Since the definition of $T_i$ depends on the choice of orthonormal basis
${\underline s}=(s_0,...,s_N)$, we shall sometimes write $T_i = T_{i,{\underline s}}$ when we want to stress
the dependence on ${\underline s}$.

\vskip .1in

 Thus we assume ${\rm Ric}(\omega_i)=-\omega_i$ and we wish
to construct imbeddings $T_i: X_i\rightarrow{\mathbb P}^N$ such that the sequence $W_i = T_i(X_i)\subseteq{\mathbb P}^N$ converges to a
singular  K\"ahler-Einstein variety $W_\infty$ with $K_{W_\infty}>0$.
\vskip .1in
The condition that $W_\infty$ is a ``singular  K\"ahler-Einstein variety" can be made precise as follows.
Let $W\subseteq{\mathbb P}^N$ be a projective variety with $K_W$ ample. The work of Berman-Guenancia \cite{BG} combined with the results of Odaka \cite{0} tell
us that the following conditions are equivalent.

\begin{enumerate}
\item There is a {\rm K\"ahler } metric $\omega$ on $W^{\rm reg}$ such that ${\rm Ric}(\omega)=-\omega$ satisfying the volume condition $\int_{W^{\rm reg}}\omega^n=c_1(K_W)^n$.
\vskip .02in
\item $W$ has at worst semi-log canonical singularities.
 \item $W$ is K-stable
\end{enumerate}

We wish to construct $T_i$ in such a way that  $W_\infty=\lim_{i\to\infty}T_i(X_i)$
has at worst semi-log canonical singularities. In this paper we  restrict
our attention to the case $n=1$.
\vskip .1in
Our long-term goal is to generalize the above theorem of \cite{DS} to the case where the $(X_i,\omega_i)$ are
smooth canonical models, of dimension $n$, i.e. $X_i$ is smooth and $c_1(X_i)<0$. The proof we present here is designed with that goal in mind. There are other approaches, but this is the one that seems to lend itself most easily to generalization. We have been able to extend the techniques to the case of dimension two, but that will be the subject a future paper.

\vskip .1in
\begin{remark}\label{fail}
One might guess, in parallel with the Fano setting, that the $T_i: X_i\rightarrow{\mathbb P}^N$ should be
the pluricanonical Bergman imbeddings, that is $T_i=T_{i,{\underline s}}$ where
${\underline s}=(s_0,...,s_N)$ and the $s_\alpha$
form an orthonormal basis of $H^0(X_i,mK_{X_i})$ with respect to the inner product (\ref{berg}). But as we shall see,
this doesn't produce the correct limit, i.e. $W_\infty$, the limiting variety, is not stable. In order to get the right imbedding into projective space, we need to replace $T_{i,{\underline s}}$ with $T_{i,{\underline s}}^\epsilon$, the so called $\epsilon$-Bergman imbedding, defined below.
\end{remark}
\vskip .1in
We first need to establish some notation. Fix $g\geq 2$ and $\epsilon>0$. If $X$ is a compact Riemann surface
of genus $g$, or more generally a stable analytic curve (i.e. a Riemann surface with nodes whose universal cover is the Poincar\'e disk) of genus $g$, we define the $\epsilon$-thick part of $X$ to be
$$ X_\epsilon\ := \ \{x\in X\,:\, {\rm inj}_x\geq\epsilon\}
$$
Here ${\rm inj}_x$ is the injectivity radius at $x$ and  the metric $\omega$ on $X$  is the unique hyperbolic metric satisfying ${\rm Ric}(\omega)=-\omega$.
It is well known that there exists $\epsilon(g)>0$ such that for all $X$  of genus $g$, and for all $0<\epsilon<\epsilon(g)$, that $X\backslash X_\epsilon$ is a finite disjoint union of holomorphic annuli.
\vskip .1in
Next we define the ``$\epsilon$-Bergman imbedding" $T_{{\underline s}}^\epsilon: X\rightarrow{\mathbb P}^N$. Fix
$0<\epsilon<\epsilon(g)$ and fix $m\geq 3$. For each stable analytic curve of genus $g$,
 we choose a basis ${\underline s}=\{s_0,...,s_{N_m}\}$ of
$H^0(X,mK_X)$
 such that
$$  \int_{X_\epsilon} (s_\alpha,s_\beta)\,\omega\ = \ \delta_{\alpha,\beta}
$$
Here
$(s_\alpha,s_\beta)=s_\alpha\bar s_\beta\omega_i^{-m}$ is the  usual pointwise inner product.
Such a basis is uniquely determined up to the action of $U(N+1)$.
Let $T_{{\underline s}}^\epsilon: X\hookrightarrow{\mathbb P}^{N_m}$ be the map $T_{\underline s}^\epsilon=(s_0,...,s_{N_m})$. Let $W=T_{\underline s}^\epsilon(X)$. One easily checks that $W$ is a stable algebraic curve and   $T_{\underline s}^\epsilon: X\rightarrow W$ is a biholomorphic map.
In particular, we have the following simple lemma.

\begin{lemma}\label{iso}
 If $X_0$ and $X_0'$ are stable analytic curves, and ${\underline s}, {\underline s}'$ are orthonormal bases for
$H^0(X_0, mK_{X_0})$ and $H^0(X_0, mK_{X_0'})$ respectively, then the
following conditions are equivalent
\begin{enumerate}
\item $X_0\ \approx \ X_0'$  (i.e. $X_0$ and $X_0'$ are biholomorphic).
\item $[T^\epsilon_{{\underline s}'}(X_0')]\ \in \ U(N+1)\cdot [T^\epsilon_{\underline s}(X_0)]$
\item $[T^\epsilon_{{\underline s}'}(X_0')]\ \in \ SL(N+1,{\mathbb C})\cdot [T^\epsilon_{\underline s}(X_0)]$
\end{enumerate}
Here $[T_{\underline s}^\epsilon X_0]\in {\rm Hilb}$  is the point representing $T_{\underline s}^\epsilon X_0\subseteq{\mathbb P}^N$ in ${\rm Hilb}$, the Hilbert scheme.
\end{lemma}
\vskip .1in

\vskip .1in
Now let $X_i$ be a sequence of stable analytic curves of genus $g$ (e.g Riemann surfaces of genus $g$). Then a basic theorem of Bers \cite{B} (we shall outline the proof below) says there exists a stable analytic curve $X_\infty$ (for a
precise definition see Definition \ref{staban}) such that after passing to a subsequence, $X_i\rightarrow X_\infty$.  By this we mean
$X_i^{\rm reg} \rightarrow X_\infty^{\rm reg}$ in the pointed Cheeger-Colding topology (see Definition \ref{CG}). Here, for $1\leq i\leq \infty$,  $X_i^{\rm reg}\subseteq X_i$ is the smooth locus.
This provides the analogue of the left vertical arrow in (\ref{diag}).

\begin{theorem}\label{main} Let $X_i$ be a sequence of stable analytic curves of genus $g$. After passing to a subsequence we have
 $X_i\rightarrow X_\infty$ in the Cheeger-Colding sense as above. Then there is a stable algebraic curve $W_\infty$ and orthonormal bases ${\underline s}_i$ of $H^0(X_i, mK_{X_i})$, such that if $W_i=T_i^\epsilon(X_i)$ then $W_i\rightarrow W_\infty$ in the algebraic sense, i.e. as points in the Hilbert scheme. Moreover,  $T_\infty|_{X_i^{\rm reg}}$  satisfies
property (\ref{Tinf}).
\end{theorem}

The idea of using Teichmuller theory  to understand moduli space was advocated by Bers \cite{B, B1, B2, B3}
 in a  of project he initiated, and  which was later completed by Hubbard-Koch \cite{HK}. They define an analytic quotient of ``Augmented Teichmuller Space" whose quotient by the mapping class group is  isomorphic to compactified moduli space as analytic spaces. Our approach is different and is concerned with the imbedding of the universal curve into projective space.
\vskip .1in
\begin{remark}. As we vary $\epsilon$, the  maps $T_i^\epsilon$ differ by uniformly bounded transformations. We shall
see that if $0<\epsilon_1,\epsilon_2<\epsilon(g)$ then $T^{\epsilon_1}_i=g_i\circ T^{\epsilon_2}_i$ where the change of
basis matrices $g_i\in GL(N+1,{\mathbb C})$ converge: $g_i\rightarrow g_\infty\in GL(N+1,{\mathbb C})$. In particular,
$\lim_i T^{\epsilon_1}_i(X_i)$ and $\lim_i T^{\epsilon_2}_i(X_i)$ are isomorphic.
\end{remark}
\vskip .1in
As a corollary of our theorem we shall give a ``metric" proof of the
stable reduction theorem due to  Deligne-Mumford \cite{DM}:
\vskip .1in
\begin{theorem}\label{1point2} Let $C$ be a smooth curve and  $f:{\mathcal X}^0\rightarrow C^0$ be a flat family of stable analytic curves over a Zariski open subset $C^0\subseteq C$. Then there exist a branched cover $\tilde C\rightarrow C$ and a flat family $\tilde f:\tilde{\mathcal X}\rightarrow \tilde C$ of stable analytic curves extending ${\mathcal X}^0\times_{\tilde C}C^0$. Moreover, the extension is unique up to finite base change.
\end{theorem}

 In addition we show that the central
fiber  can be characterized as the Cheeger-Colding limit of the general fibers.
More precisely:
\begin{proposition} Endow $X_t$ with its unique K\"ahler-Einstein metric normalized so that  ${\rm Ric}(\omega_t)=-\omega_t$. Then for every $t\in C^0$ there exist points $p_t^1,....,p_t^\mu\in X_t:= f^{-1}(t)$ such that the pointed Cheeger-Colding limits
$Y_j=\lim_{t\rightarrow 0}(X_t,p_t^j)$ are the connected components of $\tilde X_0\backslash\Sigma$ where
$\tilde X_0:=\tilde f^{-1}(0)$ and $\Sigma\subseteq \tilde X_0$ is the set of nodes of $\tilde X_0$. Moreover the limiting
metric on $X_\infty$ is its unique K\"ahler-Einstein metric.
\end{proposition}
\begin{remark} A slightly modified proof also gives the log version of stable reduction, i.e for  families $(X_t,D_t)$ where $D_t$ is an effective divisor supported on $n$ points and $K_{X_t}+D_t$ is ample. We indicate which modifications are necessary at the end of section 3.
\end{remark}
\vskip .1in
\begin{remark} In \cite{S} and \cite{SSW}, Theorems 1.1 and Corollary 2.1 are shown to hold for smooth canonical models of dimension $n>1$. But these papers assume the general version of Theorem 1.2, i.e. of stable reduction. In this paper we do not make these assumptions. In fact, our main purpose here is to prove these algebraic geometry results using analytic methods.
\end{remark}
\vskip .1in

We shall first prove Theorem 1.1 under the assumption that the $X_i$ are smooth, and Theorem 1.2 under the assumption that the generic fiber of $f$ smooth. Afterwards we will  treat the general case.

\section{Background}
Let $X$ be a compact connected Hausdorff space, let $r\geq 0$  and $\Sigma=\{z_1,...,z_r\}\subseteq X$. We say that $X$ is a nodal
analytic curve if $X\backslash \Sigma$ is a disjoint union $Y_1\cup \cdots \cup Y_\mu$ of punctured compact Riemann surfaces and if for every $z\in \Sigma$, there is a small open set $z\in U\subseteq X$ and a continuous function

$$ f: U \rightarrow \{(x,y)\in {\mathbb C}^2\,:\, xy=0\}
$$
with the properties:
\begin{enumerate}
\item $f(z)=(0,0)$
\item $f$ is a homeomorphism onto its image
\item $f|_{U\backslash \{z\}}$ is holomorphic
\end{enumerate}

\noindent
If $r=0$ then $X$ is a compact Riemann surface.

\vskip .1in
\begin{definition}\label{staban}
We say that a nodal analytic curve $X$ is a stable analytic curve if each of the $Y_j$ is covered by the Poincar\'e disk.  In other words, each of the $Y_j$ carries a unique hyperbolic metric (i.e. a metric whose curvature is $-1$) with finite volume.
\end{definition}
\vskip .1in
If $X$ is a stable analytic curve we let $K_X$ be its canonical bundle. Thus the restriction of $K_X$ to $X\backslash\Sigma$ is the usual canonical bundle. Moreover, in the neighborhood of a point $z\in\Sigma$, that is in a neighborhood of
of $\{xy=0\}\subseteq{\mathbb C}^2$,  a section of $K_X$ consists of a pair of meromorphic differential forms $\eta_1$ and $\eta_2$ defined on $x=0$ and $y=0$ respectively, with the following properties: both are holomorphic away from the origin, both have at worst simple poles at the origin, and $res(\eta_1)+res(\eta_2)=0$.
\vskip .1in
We briefly recall the proof of the above characterization of $K_X$ for nodal curves. A nodal singularity is \def{\rm Spec}{{\rm Spec}} ${\rm Spec}(B)$  where  $B={\mathbb C}[X,Y]/(Y^2-X^2)$. Then ${\mathbb C}[X] \rightarrow {\mathbb C}[X,Y]$ is generated by $Y$ which satisfies the monic equation $Y^2-X^2 = 0 $. According the Lipman's characterization of the canonical sheaf \cite{Lip} if $B=C[Y]/(f)$
where $C={\mathbb C}[X_1,...,X_n]$ and $f$ is a monic polynomial in $Y$ with coefficients in $C$, and if $X={\rm Spec}(B)$, then $K_X$ is the
sheaf of holomorphic $(n,n)$ forms on $X_{{\rm reg}}$ which can be written as $F\cdot {\pi^*(dx^1\wedge\cdots dx^n)\over f'(Y)}$
where $\pi:X\rightarrow{\rm Spec}(C)$ and $F$ is a regular function on $X$. In our case, $f(Y)=Y^2-X^2$ so $f'(Y)=2Y$ which means that
$K_X$ is free of rank one, generated by ${dx\over 2y}$ or equivalently ${dx\over y}$. If we consider the map
${\mathbb C}\rightarrow X$ given by $t\mapsto (t,t)$ then $dx\over y$ pulls back to $dt\over t$. On the other hand, if we consider
$t\mapsto (t,-t)$ then $dx\over y$ pulls back to $-{dt\over t}$.
\vskip .1in
If $X$ is a compact Riemann surface of genus $g\geq 2$, then ${\rm vol}(X)=2g-2$. If $X$ is a stable analytic curve, we say that $X$ has genus $g$ if $\sum_j{\rm vol}(Y_j)=2g-2$. Here the volumes are measured with respect to the hyperbolic metric and the $Y_j$ are the irreducible components of $X^{\rm reg}$.
\vskip .1in
Let $X$ be a stable analytic curve.
The following properties of  $K_X$ are proved in Harris-Morrison \cite{HM}:  \begin{enumerate}
\item $h^0(X,mK_X)=(2m-1)(g-1):=N_m-1$ if $m\geq 2$.

\item $mK_X$ is very ample if $m\geq 3$
\item If $m\geq 3$ the $m$-pluricanonical imbedding of $X$ is a stable algebraic curve in ${\mathbb P}^{N_m}$

\end{enumerate}\vskip .1in

Next we recall some basic results from Teichmuller theory.
Fix $g>0$ and fix  $S$, a smooth surface of genus $g$.  Teichmuller space ${\mathcal T}_g$
is the set of equivalence classes of  pairs $(X,f)$ where $X$ is a compact Riemann surface of genus $g$ and $f: S\rightarrow X$ is a diffeomorphism. Two pairs $(X_1,f_1)$ and $(X_2,f_2)$ are equivalent if there is a bi-holomorphic map $h: X_1\rightarrow X_2$ such that
$f_2^{-1}\circ h\circ f_1: S\rightarrow S$ is in ${\rm Diff}_0(S)$, diffeomorphisms isotopic to the identity. The pair $(X,f)$ is called a ``marked Riemann surface". The space ${\mathcal T}_g$ has a natural topology: A sequence $\tau_n\in{\mathcal T}_g$ converges to $\tau_\infty$ if we can
find representatives $f_n: S\rightarrow X_n$, $1\leq n\leq \infty$ such that  the sequence of diffeomorphisms
$f_\infty^{-1}\circ h\circ f_n$ converges to the identity.
\vskip .1in

The space ${\mathcal T}_g$ has a manifold structure given by Fenchel-Nielsen Coordinates whose construction we now recall. Choose a graph $\Gamma$ with the following properties: $\Gamma$ has $g$ vertices, each vertex is connected to three edges (which are not necessarily distinct since  we allow an edge to connect a vertex to itself). For example, if $g=2$, then there are two such graphs: Either $v_1$ and $v_2$ are connected by three edges, or they are connected by one edge, and each connected to itself by one edge.
\vskip .1in
Fix such a graph $\Gamma$. It has $3g-3$ edges. Fix an ordering $e_1,...,e_{3g-3}$ on the edges. Once we fix $\Gamma$ and we fix an edge ordering, we can define a map
$({\mathbb R}_{+}\times{\mathbb R})^{n} \rightarrow {\mathcal T}_g$ as follows. Given
$(l_1,\theta_1,...,l_n,\theta_n)\in {\mathbb R}^{2n}$ we associate to each vertex $v\in\Gamma$ the pair of pants whose geodesic boundary circles have lengths  $(l_i,l_j,l_k)$ where $e_i,e_j,e_k$ are the three edges emanating from $v$. Each of those circles contains two canonically defined points, which are the endpoints of the unique geodesic segment joining it to the other geodesic boundary circles.
\vskip .1in
If all the $\theta_j=0$, then we join the pants together, using the rules imposed by the graph $\Gamma$, in such a way that canonical points are identified. If some of the $\theta_j$ are non-zero, then we rotate an angle of $l_j\theta_j$ before joining the boundary curves together.
\vskip .1in
Thus we see that ${\mathcal T}_g$ is a manifold which is covered by a finite number of coordinate charts corresponding to different graphs $\Gamma$ (each
diffeomorphic to $({\mathbb R}_{+}\times{\mathbb R})^{n}$)
If we allow some of the $l_j$ to equal zero, then we can still glue the pants together as above, but this time we get a nodal curve. In this way,
$({\mathbb R}_{\geq 0}\times{\mathbb R})^{n} $ parametrizes all  stable analytic curves.
\vskip .1in

Teichmuller proved that the  manifold ${\mathcal T}_g$ has a natural complex structure, and
that there exists a universal curve ${\mathcal C}_g\rightarrow{\mathcal T}_g$, which is a map between complex manifolds, such that the fiber above $(X,f)\in {\mathcal T}_g$ is isomorphic to $X$. Moreover, if
${\mathcal X}\rightarrow B$ is any family of marked Riemann surfaces, then there exists
a unique holomorphic map $B\rightarrow {\mathcal T}_g$ such that ${\mathcal X}$ is the pullback of
${\mathcal C}_g$. Fenchel-Nielsen coordinates are compatible  with the complex structure, i.e. they  are smooth, but not holomorphic (although they are real-analytic).
\vskip .1in
\begin{remark}\label{consequence}{One consequence of Teichmuller's theorem is the following. Let
${\mathcal X}\rightarrow B$ be a holomorphic family of marked Riemann surfaces and let
$F: B\rightarrow ({\mathbb R}_+\times R)^n$ be the map that sends $t$ to the Fenchel-Nielsen coordinates of $X_t$. Then $F$ is a smooth function. In particular, $X_t\rightarrow X_0$}. This shows that in the stable reduction theorem, if a smooth fill-in exists then it is unique.
\end{remark}

Now let $X$ be a compact Riemann surface. A theorem of Bers \cite{B}, Theorem 15  (a sharp version appears in Parlier \cite{P}, Theorem 1.1) says that for $g\geq 2$ there exists a constant $C(g)$, now known as the Bers constant, with the following property. For every Riemann surface $X$ of genus $g$ there exists a representative $\tau=(X,f)\in{\mathcal T}_g$ and a graph $\Gamma$ (i.e. a coordinate chart) such that the Fenchel-Nielsen coordinates of $\tau$ are all bounded above by $C(g)$. This is analogous to the fact that  ${\mathbb P}^N$ is covered
by $N+1$ coordinate charts, each biholomorphic to ${\mathbb C}^N$, and that give a point $x\in{\mathbb P}^N$ we can choose a coordinate
chart so that $x\in {\mathbb C}^N$ has the property $|x_j|\leq 1$ for all $j$. In particular, this proves ${\mathbb P}^N$ is sequentially compact.

\vskip .1in

\vskip .1in
Bers \cite{B} uses the existence of the Bers constant to show that the space of stable analytic curves is compact with respect to a natural topology (equivalent to the Cheeger-Colding topology). For the convenience of the reader, we recall the short argument.
Let $X_j$ be a sequence of Riemann surfaces. Then after passing to a subsequence, there is a graph $\Gamma$ and representatives $\tau_j=(X_j,f_j)\in{\mathcal T}_g$  such that the Fenchel-Nielsen coordinates of $\tau_j$ with respect to $\Gamma$ are all bounded above by $C(g)$ (this is due to the fact that there are only finite many allowable graphs). After passing to a further subsequence, we see  $\tau_j\rightarrow\tau_\infty\in ({\mathbb R}_{\geq 0}\times{\mathbb R})^{n} $. If
$\tau_\infty\in ({\mathbb R}_{+}\times{\mathbb R})^{n}$ then the limit is a smooth Riemann surface. Otherwise, it is a stable analytic curve $X_\infty$. Thus

\beta\label{union}
X_\infty= \cup _{i=1}^\mu X^\alpha,\ \ {\rm and} \ \ X^{\rm reg}_\infty=\sqcup_{\alpha=1}^\mu Y^\alpha
\ee
where the second union is disjoint, and  $Y^\alpha = X^\alpha\backslash F^\alpha$ where $ X^\alpha$ is a compact Riemann surface and $F^\alpha\subseteq X^\alpha$ a finite set, consisting of the cusps.
\vskip .1in

\begin{corollary}\label{cor1} Let $p_\infty^\alpha\in Y^\alpha$. Then there exist $p_i^1,....,p_i^\mu\in X_i$ such that in the pointed Cheeger-Colding topology,
$(Y^\alpha,p_\infty^\alpha)=\lim_{j\to\infty}(X_j,p_j^\alpha)$. Moreover, for every open set $p_\infty^\alpha\in U_\infty^\alpha\subseteq\sub Y^\alpha$ there
exist open sets $p_i^\alpha\subseteq U_i^\alpha\subseteq X_i$ and diffeomorphisms $f_j^\alpha: U_\infty^\alpha\rightarrow U_j^\alpha$ so that $(f_j^\alpha)^*\omega_j^\alpha \rightarrow \omega_\infty^\alpha$ and $(f_j^\alpha)^*J_j^\alpha\rightarrow J^\alpha_\infty$ where $\omega_j^\alpha$ and $\omega_\infty^\alpha$ are the hyperbolic metrics on $U^\alpha_j$ and $U_\infty^\alpha$, and
$J_j^\alpha$ and $J_\infty^\alpha$ are the complex structures on $U^\alpha_j$ and $U_\infty^\alpha$
\end{corollary}
\vskip .1in
\begin{definition}\label{CG} In the notation of Corollary \ref{cor1},
we shall say $\omega_j\rightarrow\omega_\infty$ in the pointed Cheeger-Colding sense and we shall write $X_i\rightarrow X_\infty$.
\end{definition}
\vskip .1in
\noindent
Remark:  Odaka \cite{O2} uses pants decompositions to construct a ``tropical compactification" of moduli space which attaches metrized graphs (of one real dimension) to the boundary of moduli space. These interesting compactifications are compact Hausdorff topological spaces but are no longer algebraic varieties.

\section{Limits of Bergman imbeddings}

\vskip .1in
Now let ${\mathcal X}$ be as in the theorem, and let $t_i\in C^0$ with $t_i\rightarrow 0$. Let $X_i=X_{t_i}$  and fix a pants decomposition of $X_i$.
Then Bers' theorem implies that after passing to a subsequence we can find a nodal curve $X_\infty$ as above so that
$X_j\rightarrow X_\infty$.
\vskip .1in

In order to prove the theorem, we must show:
\begin{enumerate}
\item  $X_\infty$ is independent of the choice of subsequence.
\item  After making a finite base change, we can insert $X_\infty$ as the central fiber in such a way that the completed family is algebraic.

\end{enumerate}

We begin with (2). Let $X$ be a hyperbolic Riemann surface with finite area (i.e. possibly not compact, but only cusps). The Margulis ``thin-thick decomposition" says that there exists $\epsilon(g)>0$ with the following property. There exists at most $3g-3$ closed geodesics of length less that $\epsilon(g)$. Moreover, for every $\epsilon\leq\epsilon(g)$ the set

$$ {X\backslash X_\epsilon}\ := \ \{x\in X\,:\, {\rm inj}_x<\epsilon\}
$$
is a finite union of of holomorphic annuli (which are open neighborhoods of short geodesics) if $X$ is compact, and a finite union of annuli as well as punctured disks, which correspond to cusp neighborhoods if $X$ is has singularities. We call these annuli ``Margulis annuli".
Moreover,  $V(\epsilon)$, the volume of $X\backslash X_\epsilon$, has the property $\lim_{\epsilon\rightarrow 0}V(\epsilon)=0$. An elementary proof is given in Proposition 52, Chapter 14 of Donaldson \cite{D}.
\vskip .1in

Now we define a modified Bergman kernel as follows: For convenience we write $\epsilon=\epsilon(g)$. This is a positive constant, depending only on the genus $g$. Let $X$ be a stable analytic curve. For
$\eta_1,\eta_2\in H^0(X,mK_X)$ let

\beta\label{berg1} \<\eta_1,\eta_2\>_\epsilon\ = \ \int_{X_\epsilon} \eta_1\bar\eta_2 h_{KE}^m \omega_{KE}
\ee
\noindent
and $\|\eta\|_\epsilon^2=\<\eta,\eta\>_\epsilon$.
If we replace $X_\epsilon$ by $X$, we  get the standard Bergman inner product.

\vskip .1in

\vskip .1in

Now fix $m\geq 3$. Choosing orthonormal bases with respect to the inner product (\ref{berg1}) defines imbeddings $T_i^\epsilon: X_i\rightarrow {\mathbb P}^{N_m}$ and $T_\infty^\epsilon: X_\infty\rightarrow{\mathbb P}^{N_m}$, which we call $\epsilon$-Bergman imbeddings.
Our goal is to show
\begin{theorem}\label{Tilimit} Let $X_1, X_2, ... $ be a sequence of compact Riemann surfaces of genus $g$. Then there exists a stable analytic curve $X_\infty$ such that after passing to a subsequence, $X_i\rightarrow X_\infty$ in the Cheeger-Colding topology.  For $1\leq i<\infty$, we fix an orthonormal basis ${\underline s}_i$ of $H^0(X_i, mK_X)$. Then there exists a choice of orthonormal basis ${\underline s}_\infty$ for $X_\infty$ such that
after passing to a subsequence,
\beta\label{limit}  \lim_{i\to\infty}T_{i,{\underline s}_i}^\epsilon\ = \ T_{\infty,{\underline s}_\infty}^\epsilon
\ee
In other words, if $x_i\in X_i$ and $x_\infty\in X_\infty$ with $x_i\rightarrow x_\infty$, then  $$T_i^\epsilon(x_i)\rightarrow T_\infty^\epsilon(x_\infty)$$
\end{theorem}\noindent
The proof of Theorem \ref{Tilimit} rests upon the following.
\vskip .1in
\begin{theorem}\label{ee2} Fix  $g\geq 2$  and $m,\epsilon>0$. Then there exist $C(g,m,\epsilon)$ with
the following property.
$$ \|s\|_\epsilon\ \leq \ \|s\|_{\epsilon/2}\ \leq \ C(g, m,\epsilon)\|s\|_\epsilon
$$
for all Riemann surfaces $X$ of genus $g$ and all $s\in H^0(X,mK_X)$.
\end{theorem}

To prove the theorem, we need the following adapted version of a
result of Donaldson-Sun. We omit the proof which is very similar to \cite{DS} (actually easier since the only singularities of $X_\infty$ are nodes so the pointed limit of the $X_i$ in the Cheeger-Colding topology is smooth).

\begin{proposition}\label{propDS}
Let $X_i\rightarrow X_\infty$ be a sequence of Riemann surfaces of genus $g$ converging in the pointed Cheeger-Colding sense to a stable curve $X_\infty$.
Fix
$\{s_0^\infty,...,s_M^\infty\}\ \subseteq\
H^0(X_\infty,mK_{X_\infty})$
 an $\epsilon$-orthonormal basis   of the bounded sections (i.e. the $L^2(X_\infty)$ sections, i.e. the sections which vanish at all nodes). Then there exists an $\epsilon$-orthonormal subset
$$\{s_0^i,...,s_M^i\}\ \subseteq\
H^0(X_i,mK_{X_i})$$
such that for $0\leq \alpha\leq M$, we have
$$s_\alpha^i\rightarrow s_\alpha^\infty
 $$
in $ L^2$
and uniformly on compact subsets of $X_\infty^{\rm reg}$. In particular, if $x_i\in X_i^{\rm reg}$

\beta\label{DS}
\hbox{
$x_i\rightarrow x_\infty \iff
s_\alpha^i(x_i)\rightarrow s_\alpha^\infty(x_\infty)\ \ {\rm for\ all}\ \ 0\leq\alpha\leq M
\ \ \iff\ \ T^{\nu,\epsilon}_i(x_i)\rightarrow T^{\nu,\epsilon}_\infty(x_\infty)$ .
}
\ee
where $T^{\nu,\epsilon}_i: X_i^{\rm reg}\hookrightarrow {\mathbb P}^M$ is the map $x_i\mapsto
(s_0^i,...,s_M^i)(x_i)$ for $1\leq i\leq \infty$.

\end{proposition}

\vskip .1in
{\it Proof of theorem \ref{ee2}.} Let $X_i\rightarrow X_\infty$ as in Proposition \ref{propDS}. Choose $(s_0^\infty,...,s_M^\infty)$ and
$(t_0^\infty,...,t_M^\infty)$ which are $\epsilon$ and $\epsilon/2$ orthonormal bases of the subspace of bounded sections in $H^0(X_\infty, mK_{X_\infty})$ in such a way that
$t_\alpha^\infty=\lambda_\alpha^\infty s_\alpha^\infty$ for real numbers $0<\lambda_\alpha^\infty<1$. Choose
$s_\alpha^i\rightarrow s_\alpha^\infty$ and $t_\alpha^i\rightarrow t_\alpha^\infty$ as in Proposition \ref{propDS} in such a way that $t_\alpha^i=\lambda_\alpha^is_\alpha^i$ with
$0<\lambda_\alpha^i<1$. Clearly
\beta\label{22} \lambda_\alpha^i \ \rightarrow \ \lambda_\alpha^\infty>0\ \ {\rm for} \ \ 0\leq \alpha\leq M
\ee
Choose additional sections $s_\alpha^i$ and $t_\alpha^i$ for $M+1\leq \alpha\leq N$
so that $\{s_0^i,..., s_N^i\}$ and $\{t_0^i,..., t_N^i\}$ are $\epsilon$ and $\epsilon/2$ bases of $H^0(X_i,mK_{X_i})$ and $t_\alpha^i=\lambda_\alpha^is_\alpha^i$ with
$0<\lambda_\alpha^i<1$ for $0\leq\alpha\leq N$.
\vskip .1in
Now assume the theorem is false. Then there exists $X_i\rightarrow X_\infty$ as above such that $\lambda_\alpha^i\rightarrow 0$ for some $\alpha$.
 We must have $\alpha\geq M+1$ by (\ref{22}).
Choose $M+1\leq A<N$ such that $\lambda_\alpha^i\rightarrow 0$ if and only if $A\leq \alpha\leq N$.
\vskip .1in\noindent
Since $\|s_\alpha^i\|_{L^2(X_\epsilon)}=1$ we may choose $s_\alpha^\infty(\epsilon)\in H^0(X_i^\epsilon, K_{X_\infty}|_{X_\infty^\epsilon})$ such that
\beta s_\alpha^i|_{X^\epsilon_i} \rightarrow s_\alpha^\infty(\epsilon)\ \ {\rm for}\ \ M+1\leq \alpha\leq N
\hbox{\ \ uniformly on compact subsets of $X_\epsilon$}
\ee

Let $T_i^\epsilon: X_i\rightarrow W_i^\epsilon\subseteq{\mathbb P}^N$ be the Kodaira map given by the
sections $s_0^i,...,s_N^i$ and let {$W_\infty^\epsilon=\lim_{i\to\infty}W_i^\epsilon$. }
Let
$$ T_\infty^\epsilon: X_\infty^\epsilon \hookrightarrow W_\infty^\epsilon \ \ {\rm and} \ \
T_\infty^{\nu,\epsilon}: X^{\rm reg}_\infty \hookrightarrow{\mathbb P}^M
$$
be the Kodaira maps given by $(s_0^\infty,...,s_M^\infty, s_{M+1}^\infty(\epsilon),...,s_N^\infty(\epsilon))$ and $(s_0^\infty,...,s_M^\infty)$.  Thus
\beta\label{inc} \pi\circ T_\infty^\epsilon\ = \ T_\infty^{\nu,\epsilon}|_{X_\infty^{\epsilon}}
\ee
where
$\pi:{\mathbb P}^N_M:={\mathbb P}^N\backslash\{z_0=\cdots = z_M=0\}\rightarrow {\mathbb P}^M$ is
defined by $(z_0,...,z_N)\mapsto (z_0,...,z_M)$. Moreover
$$ \pi(W^\epsilon_\infty\cap{\mathbb P}^N_M)\ \subseteq \ T_\infty^{\nu,\epsilon}(X_\infty^{\rm reg})
$$

\vskip .1in
Now the definition of $A$ implies
$$  T_\infty^{\epsilon/2}(X^{\epsilon}_{\infty})\ \subseteq\ Z^{\epsilon/2}_\infty\ := \ \{z\in  W^{\epsilon/2}_\infty\,:\, z_{A}=z_{A+1}=\cdots=z_N=0\}
$$
Thus (\ref{inc})  implies
$$ T_\infty^{\nu,\epsilon/2}(X_\infty^{\rm reg})\ \supset\
\pi(Z^{\epsilon/2}_\infty\cap {\mathbb P}^N_M)\supset \pi(T_\infty^{\epsilon/2}(X^{\epsilon}_{\infty}))\ = \
T_\infty^{\nu,\epsilon/2}(X_\infty^\epsilon)
$$
Since the second set is constructible,
$$ \pi(Z^{\epsilon/2}_\infty\cap {\mathbb P}^N_M) \ = \  T_\infty^{\nu,\epsilon/2}(X_\infty^{\rm reg}\backslash \Sigma_\epsilon)
$$
where $\Sigma_\epsilon\subseteq X_\infty^{\rm reg}\backslash X_\infty^\epsilon$ is a finite set. Moreover, $\Sigma_\epsilon$ is monotone in $\epsilon$.
\vskip .1in
Let $x_\infty\in X_\infty^{\rm reg}\backslash\Sigma_\epsilon$. Then $T_\infty^{\nu,\epsilon/2}(x_\infty) = \pi(w_\infty)$
for some $w_\infty\in Z_\infty^{\epsilon/2}\cap{\mathbb P}^N_M$. Choose $w_i\in W_i^{\epsilon/2}$ such
that $w_i\rightarrow w_\infty$ and choose $x_i\in X_i$ such that $T_i^{\epsilon/2}(x_i)=w_i$.
Then (\ref{DS}) implies

$$ T^{\epsilon/2}_i(x_i)\rightarrow w_\infty\ \Longrightarrow\ \pi(T^{\epsilon/2}_i(x_i))\rightarrow \pi(w_\infty)\ \Longrightarrow \ T_i^{\nu,\epsilon/2}(x_i)\ \rightarrow \ T_\infty^{\nu,\epsilon/2}(x_\infty) \ \Longrightarrow \ x_i\rightarrow x_\infty
$$

Thus we see that if $x_\infty\in X_\infty^{\rm reg}\backslash\Sigma_\epsilon$ there exists
$x_i\rightarrow x_\infty$ such that

$$ \lim_{i\to\infty} T_i^{\epsilon/2}(x_i) \ \in \ Z_\infty^{\epsilon/2}
$$
\begin{lemma}\label{Sigma} If $\epsilon_1<\epsilon_2$ then $\Sigma_{\epsilon_1}\subseteq \Sigma_{\epsilon_2}\cap X_\infty\backslash X_\infty^{\epsilon/2}$.
\end{lemma}
{\it Proof.} Assume $x_\infty\notin \Sigma_{\epsilon_2}$. Then there exists $x_i\rightarrow x_\infty$ such that
$ \lim_{i\to\infty}s^i_\alpha(x_i) =0
$
for all $A\leq \alpha\leq N$. Then
$ \lim_{i\to\infty}t^i_\alpha(x_i) =\lim_{i\to\infty}\lambda_\alpha^is^i_\alpha(x_i) =0
$
since $0<\lambda_\alpha^i<1$. \qed
\vskip .1in
We may there assume, after possibly decreasing $\epsilon$, that $\Sigma_\epsilon=\emptyset$.  This means that for every $x_\infty\in X_\infty^{\rm reg}$
there exists $x_i\rightarrow x_\infty$ such that
$ \lim_{i\to\infty}s^i_\alpha(x_i) =0
$
for all $A\leq \alpha\leq N$.
\vskip .1in
Let $x_\infty\in X_\infty^{\rm reg}$. We say that $x_\infty$ is an \emph{$\epsilon$-good} point if
for every $x_i\rightarrow x_\infty$, $ \lim_{i\to\infty}s^i_\alpha(x_i) =0
$
for all $A\leq \alpha\leq N$. The set of $\epsilon$-bad points is finite (otherwise
$W^\epsilon_\infty$ would have infinitely many components {by the intermediate value theorem}). Also, every point in $X_\infty^{2\epsilon}$ is $\epsilon$-good.

\vskip .1in

\begin{lemma}\label{good} If $x_\infty$ is $\epsilon_2$-good then it is
$\epsilon_1$-good for all $\epsilon_1<\epsilon_2$.
\end{lemma}
{\it Proof.} Assume $x_\infty$ is $\epsilon_2$ good, let $x_i\rightarrow x_\infty$ and let $\epsilon_1<\epsilon_2$. Then
$ \lim_{i\to\infty}t^i_\alpha(x_i) =\lim_{i\to\infty}\lambda_\alpha^is^i_\alpha(x_i) =0
$
since $0<\lambda_\alpha^i<1$. \qed
\vskip .1in
Lemma \ref{good} implies that by decreasing $\epsilon$ if necessary, that all points $x_\infty\in X_\infty^{\rm reg}$ are $\epsilon$-good. But $\|s_A^i\|_\epsilon=1$ so there exists $x_i\in X_i^\epsilon$ such that $|s_A^i(x_i)|=1$. After passing to a subsequence, $x_i\rightarrow x_\infty\in X_\infty^\epsilon$ but
$\lim_i|s_A^i(x_i)|=1\not=0$ a contradiction.

\vskip .1in

We conclude that if $\eta_j\in H^0(X_j, mK_{X_j})$ is a sequence
such that the norms $\|\eta_j\|^2_\epsilon=\<\eta_j,\eta_j\>_\epsilon =1$, then after passing to a subsequence, we have $(f^\alpha_j)^*\eta_j \rightarrow  \eta_\infty$ for some $\eta_\infty\in H^0(X_\infty^{reg}, mK_{X_\infty}|_{X_\infty^{\rm reg}})$ with $\|\eta\|_\epsilon=1$. Here the $f_j^\alpha:U^\alpha_j\rightarrow U^\alpha$ are as in the statement of Corollary 1 and this is true for all $U^\alpha$ and all $\alpha$.  Moreover, an
orthornormal basis of $H^0(X_j, mK_{X_j})$, which is a vector space of dimension $(2m-1)(g-1)$,  will converge to an orthonormal set of $(2m-1)(g-1)$ elements in
$H^0(X_\infty^{reg},mK_{X_\infty})$. The main problem is to now show that these $(2m-1)(g-1)$ elements extend to elements of $H^0(X_\infty,mK_{X_\infty})$. If they extend, then they automatically form a basis since
$H^0(X_\infty,mK_{X_\infty})$ has dimension $(2m-1)(g-1)$ and this would prove
Theorem \ref{Tilimit}.
\vskip .1in

\vskip .1in
To proceed, we make use of the discussion of the Margulis collar  in  section 14.4.1 of \cite{D}.  Let $\lambda>0$ be the length of $C$ a collapsing geodesic in $X_j$ which forms a node in the limit in $X_\infty$. We fix $j$ and we write $X=X_j$. Let

\vskip .1in
$$ A_\lambda\ = \ \{z\in{\mathbb C}: 1\leq |z|\leq e^{2\pi\lambda},\, \lambda\leq \arg(z)\leq \pi-\lambda\,\}/\sim
$$
where the equivalence relation identifies the circles $|z|=1$ and $|z|=e^{2\pi\lambda}$.
Then \cite{D} shows $A$ injects holomorphically into $X$ in such a way that $1\leq y\leq e^{2\pi\lambda}$
maps to $C$. The point is that the segment $1\leq y\leq e^{2\pi\lambda}$ is very short - it has size $\lambda$. But the segments $A\cap \{\arg(z)=\lambda\}$ and $A\cap \{\arg(z)=\pi-\lambda\}$ have size 1. So for $\lambda$ small, $A$ is a topologically a cylinder, but metrically very long and narrow in the middle but not narrow at the ends.  In other words, the middle of $A$ is in the thin part, but the boundary curves are in the thick part.

\vskip .1in

 The transformation

$$ \tau\ = \ \exp\big(i{\ln z\over \lambda}\big)
$$
maps $A_\lambda$ to the annulus

$$ A'_\lambda \ = \ \{\exp(-(\pi-\lambda)/\lambda)\leq |\tau|\leq \exp(-1)\,\}
$$

To summarize: We are given a sequence $X_j$, and a geodesic $C_j$ in $X_j$ that collapses to a node $\nu$ in $Y^\alpha$ for some $\alpha$. We are also given a sequence of orthonormal bases
$\{{\eta_{j,1}},..., \eta_{j,N}\}$ of
$H^0(X_j,kK_{X_j})$ where $N=(2k-1)(g-1)$ and $\eta_{j,\mu}\rightarrow \eta_{\infty,\mu}$. Here $\eta_{\infty,\mu}$ is a section of $kK_{X\infty}$ on $X_\infty^{\rm reg}$. Fix $\mu$ and write $\eta_j=\eta_{j,\mu}$ and $\eta_\infty=\eta_{\infty,\mu}$. We need to show that $\eta_\infty$ extends to all of $X_\infty$.
\vskip .1in
We may view $\eta_j$ as a $k$ form on $A_{\lambda_j}$ or on $A'_{\lambda_j}$ and $\eta_\infty$ as a $k$ form on the punctured disk $A'_0$. Write
$\eta_j = f_j(z)dz^k=h_j(\tau)d\tau^k$ and $\eta_\infty=h_\infty(\tau)d\tau^k$. The discussion in \cite{D} shows that if we fix a relatively compact open subset  $U\subseteq A'_0$, then $h_j \rightarrow h_\infty$ uniformly on $U$ (this makes sense since $U\subseteq A'_{\lambda_j}$ for $j$ sufficiently large).

\vskip .1in
Since $\|\eta_j\|_{L^2}=1$  we have uniform sup norm bounds on the thick part of $X_j$. Thus

\beta\label{thick} \|\eta_j\|_{L^\infty((X_i)_\epsilon}\ \leq \ C(\epsilon)
\ee

We want to use (\ref{thick}) to get a bound on the thin part. In $z$ coordinates, (\ref{thick}) implies

\beta\label{sup}  |\eta|_{\omega}\ = \ |{\rm Im}(z)|^k\cdot |f(z)|\ \leq \ C(\epsilon) \ \ \hbox{if $\arg(z)=\lambda$ or $\arg(z)=2\pi-\lambda$}
\ee
since the boundary curves $\arg(z)=\lambda$ and $\arg(z)=2\pi-\lambda$ are in the thick part.
Here we write $\eta$ for $\eta_j$ and $f$ for $f_j$ to lighten the notation.
\vskip .1in
Now

\beta\label{23}{\rm Im}(z)\ = \ -\exp(\lambda\arg\tau)(\sin(\lambda\ln|\tau|)\ee

if we write $f(z)=g(\tau)$, then (\ref{sup}) implies

\beta |g(\tau)|\leq {C(\epsilon)\over\lambda^k}\ \ \hbox{for $\tau\in\partial A'$}
\ee

Since $f(z)dz^k=h(\tau)d\tau^k=g(\tau)({dz\over d\tau})^k\,d\tau^k$ and
${dz\over d\tau}= {z\lambda\over i\tau}$
we see  for $\lambda$ small

$$ |h_j(\tau)|\ \leq {1\over \lambda^k}\bigg|{dz\over d\tau}\bigg|^k\ = \
{1\over\lambda^k}{|z|^k\lambda^k\over|\tau|^k}\ \leq \ {2\over |\tau|^k}
$$
where the last inequality follows from the fact $1\leq |z|\leq 2$.
Writing
$$ u_j(\tau)\ :=\ h_j(\tau)\tau^k
$$
Thus we see $|u_j(\tau)|\leq 2$ for $\tau\in\partial A'$. The maximum principle now implies that $ |u_j(\tau)|\leq 2$ for $\tau\in A'$. Since this is true for all
$X_i$, we see that any limit $u_\infty$ must satisfy the same inequality in the limit of the annuli, which is a punctured disk: $|h_\infty(\tau)|\cdot |\tau|^k\leq C$. This shows $h_\infty$ has at most a pole of order $k$.

Moreover  $u(0)$ is
the residue \beta\label{res} u(0)\ = \ \lim_{j\to\infty}{1\over 2\pi {{\sqrt{{\scalebox{0.75}[1.0]{\( - 1\)}}}}}\hskip .01in}\int_{|\tau|=r} u_j(\tau)\,{d\tau\over\tau}
\ee
Here $0<r\leq \exp(-1)$ is any fixed number (independent of $j$).

\vskip .1in
To summarize, we have now seen that a collar degenerates to a union of two punctured disks and so the limit of
the $\eta_j$ is a pair of $k$ forms, $\eta_\infty = u_\infty(\tau)\left({d\tau\over \tau}\right)^k$ and
$\tilde\eta_\infty = \tilde u_\infty(\tau')\left({d\tau'\over \tau'}\right)^k$ where $u$ and $\tilde u$ are holomorphic
in a neighborhood of the origin in ${\mathbb C}$. There is one final condition that we need to check in order
to verify that the limit is in $H^0(X_\infty,kK_{X_\infty})$:
 Let $R=\exp(-1)$, $r=\exp(-\pi/2\lambda_j)$ and $\epsilon= \exp(-\pi/\lambda_j)$ (so $\epsilon_{\lambda_j}/r = r$).  We must show $\tilde u(0)=(-1)^ku(0)$.
\begin{center}
\begin{figure}[h!]
\begin{tikzpicture}
\coordinate (O) at (0,0);

\begin{scope}[xshift=6cm]
\coordinate (O) at (0,0);
\draw[fill=blue!25] (O) circle (2.8);
\draw[fill=red!25] (O) circle (1.2);
\draw[fill=white!25] (O) circle (0.4);

\node[text width=2cm] at (.1,-.65) {\tiny\color{black} ${\epsilon/R}\leq |\tau|\leq r$};
\node[text width=2cm] at (.35, 3.05) {\tiny\color{black} $|\tau|= R$};
\node[text width=2cm] at (.2, -2) {\tiny\color{black} $r\leq |\tau|\leq R$};

\end{scope}
\end{tikzpicture}
\caption{$A_\lambda'$}\label{X0}
\end{figure}
\end{center}

\vskip .1in
\noindent
(Here the inner white disk is $|\tau|\leq \epsilon/R$).
To check this, let $\tilde\tau= {\epsilon_j\over\tau}$. Then Figure \ref{X0} remains the same, with $\tau$ replaced by $\tilde\tau$ and
$$ f(z)dz^k\ = \ u_j(\tau)\left({d\tau\over\tau}\right)^k\ = \ u_j(\epsilon_j/\tilde\tau)(-1)^k\left({d\tilde\tau\over\tilde\tau}\right)^k
\ := \ \tilde u_j(\tilde\tau)\left({d\tilde\tau\over\tilde\tau}\right)^k
$$

Now we see

$$
\int_{|\tau|=r} u_j(\tau)\,{d\tau\over\tau}\ = \
(-1)\int_{|\tilde\tau|=\epsilon_j/r} u_j({\epsilon_j\over\tilde\tau})\,(-1){d\tilde\tau\over\tilde\tau}\ = \
(-1)^k\int_{|\tilde\tau|=r} \tilde u_j(\tilde\tau)\,{d\tilde\tau\over\tilde\tau}
$$
In the second integral, the factor of $(-1)$ outside the integral is due to the fact that the orientation of the
circle has been reversed and the $(-1)$ inside the integral comes from the change of variables. The second identity
is a result of the fact that $u(\tilde\tau)$ is holomorphic on the annulus $\{\tilde\tau\in{\mathbb C}:\epsilon_j/r < \tilde\tau< r\}$.
Taking limits as $j\to\infty$ we obtain $\tilde u(0)=(-1)^ku(0)$. This establishes Theorem \ref{Tilimit} when the $X_i$ are smooth.

\vskip .1in
Now assume the $X_i$ are stable analytic curves, but not necessarily smooth. The Fenchel-Nielsen coordinates of $X_i$ determine a point
$[X_i]\in ({\mathbb R}_{\geq 0}\times{\mathbb R})^{n} $. The simple observation we need is that $({\mathbb R}_{> 0}\times{\mathbb R})^{n}\subseteq ({\mathbb R}_{\geq 0}\times{\mathbb R})^{n} $ is dense
so we may choose a smooth Riemann surface $\tilde X_i$ such that $[X_i]
\in ({\mathbb R}_{\geq 0}\times{\mathbb R})^{n} $ is $\epsilon_i$  close to $[X_i]$ where $\epsilon_i\rightarrow 0$ (i.e. $X_i$ is smoothable). Now Corollary \ref{cor1} implies that after passing to a subsequence,
$\tilde X_i\rightarrow X_\infty$ in the pointed Cheeger-Colding topology. We conclude
that $X_i\rightarrow X_\infty$ as well. Moreover, one easily sees that $T_i^\epsilon$ and
$\tilde T_i^\epsilon$ have the same limit. This proves
(\ref{limit}) and
completes the proof of Theorem \ref{Tilimit}
 \qed
\vskip .1in
\begin{remark} The proof of the log version Theorem \ref{Tilimit} is almost the same. The only observation  we need is  the following. If $X$ is a compact Riemann surface and $D=p_1+\cdots+p_n$ is a divisor supported on $n$ points such that $K_X+D$ is ample, then $X\backslash D$ has a unique metric $\omega$ such that ${\rm Ric}(\omega)=-\omega$ and $\omega$ has cusp singularities at the points $p_j$. Morover, just as in the case $n=0$, $X$ has a pants decomposition. The only difference is that we allow some of the length
parameters to vanish, but this doesn't affect the arguments. In particular, we can use the Fenchel-Nielson coordinates to find a limit of the $(X_j,D_j)$ (after passing to a subsequence) and the $T_j^\epsilon$ are defined exactly as before.
\end{remark}
\vskip .1in
Now suppose $X_i$ is a sequence of compact Riemann surfaces of genus $g$ converging analytically to a nodal curve $X_\infty$
and let $\eta_i$ be a {\rm K\"ahler } metric on $X_i$ is the same class as the {\rm K\"ahler-Einstein\ } metric $\omega_i$. We have seen that $\omega_i\rightarrow\omega_\infty$,
the {\rm K\"ahler-Einstein\ } metric on~$X_\infty$, in the pointed Cheeger-Colding sense. Let
$\tilde\omega_\infty$
be a {\rm K\"ahler } metric on $X_\infty^{{\rm reg}}$ and assume
$\tilde\omega_i\rightarrow \tilde\omega_\infty$ in the pointed Cheeger-Colding sense. Let $T_i(\tilde\omega_i): X_i\rightarrow {\mathbb P}^N$ be the embedding defined by an orthonormal basis of $H^0(X_i,3K_{X_i})$ using the metric $\tilde\omega_i$ on the thick part of $X_i$
and define $T_\infty(\tilde\omega_\infty):X_\infty\rightarrow {\mathbb P}^N$ similarly. Thus the $T_i$ and $T_\infty$ of Theorem 2 can be written as
$T_i(\omega_i)$ and $T_\infty(\omega_\infty)$ and in this notation, Theorem 2 says
$T_i(\omega_i)\rightarrow T_\infty(\omega_\infty)$
\begin{corollary} After passing to a subsequence
$$ T_i(\tilde\omega_i)\ \rightarrow \ T_\infty(\tilde\omega_\infty)
$$
\end{corollary}

{\it Proof.} Since $\tilde\omega_\infty$ and $\omega_\infty$ are equivalent on the thick part of $X_\infty$, we see that
$$T_i(\tilde\omega_i)=\gamma_i\circ T_i(\omega_i)$$
where $\gamma_i\in GL(N+1,{\mathbb C})$ has uniformly bounded entries as does
$\gamma_i^{-1}$. Thus
after passing to a subsequence, $\gamma_i\rightarrow\gamma_\infty\in GL(N+1,{\mathbb C})$ and
$$\lim_{i\to\infty} T_i(\tilde\omega_i)\ = \ \lim_{i\to\infty}\gamma_i\circ T_i(\omega_\infty)\ = \ \gamma_\infty\circ T_\infty(\omega_\infty)\ = \  T_\infty(\tilde\omega_\infty)
$$
\noindent
Remark: The proof shows we only need to assume $\tilde\omega_i\rightarrow \tilde\omega_\infty$ on the thick part of $X_\infty$.

\section{Existence of stable fill-in} \setcounter{equation}{0}
\vskip .1in
\noindent
{\it Proof.}
Let $f:{\mathcal X}^0\rightarrow C^0=C\backslash\{p_1,...,p_m\}$ be a flat family of stable analytic curves of genus $g\geq 2$. We shall assume the fibers are smooth since except for some additional noation, the general case is proved in exactly the same way.  We first observe that we can find some completion (not necessarily nodal)
${\mathcal Y}\rightarrow C$ of the family ${\mathcal X}^0\rightarrow C^0$. To see
this let $\Omega_{{\mathcal X}^0/C^0}$ be the sheaf of relative differential forms (i.e. the relative canonical line bundle when ${\mathcal X}^0$ is smooth). Then the Hodge
bundle $f_* K_{{\mathcal X}^0/C^0}$ is a vector bundle over $C^0$ of rank $3g-3$ (see page 694 of Vakil [V])
and $f_*K_{{\mathcal X}^0/C^0}^{\otimes m}$ is a vector bundle ${\mathcal E}_m^0$ of rank $N_m-1:=(2m-1)(g-1)$ for $m\geq 2$.
Choose ${\mathcal E}_m\rightarrow C$ an extension of the vector bundle  ${\mathcal E}_m^0\rightarrow C^0$ to the curve $C$.
\vskip .1in
For example, let $U\subseteq C^0$ be any affine open subset over which ${\mathcal E}^0_m$ is trivial and let $s_0,...,s_{N_m}$ be a fixed ${\mathcal O}(U)$ basis. Then if $p_j\in V\subseteq C^0$ is an affine open set such that $V\backslash \{p_j\}\subseteq U$, then define ${\mathcal E}(V)$ to be the ${\mathcal O}(V)$ submodule of
${\mathcal E}^0(V\backslash\{p_j\})$ spanned by
the $s_\alpha$.
\vskip .1in
 Once ${\mathcal E}$ is fixed, we choose $m\geq 3$ and let ${\mathcal X}^0\hookrightarrow {\mathbb P}({\mathcal E}^0)\subseteq{\mathbb P}({\mathcal E})$ be the canonical imbedding. Then we define
 \beta\label{fake} {\mathcal Y}\subseteq {\mathbb P}({\mathcal E})
 \ee
  to be the flat limit of
${\mathcal X}^0\rightarrow C^0$ inside ${\mathbb P}({\mathcal E})\rightarrow C$.

\vskip .1in
Now we prove Theorem 1. To lighten the notation, we shall assume $m=1$ and write $C^0=C\backslash\{0\}$ where $0:= p_1$.
Suppose $t_i\in C^0$ with $t_i\rightarrow 0$ and such that we have analytic convergence  $X_{t_i}\rightarrow X_\infty$ where $X_\infty$ is an stable analytic curve.
 We wish to show that there exists a smooth curve $\tilde C$ and a finite cover $\mu:\tilde C\rightarrow C$ with the following property. If we let $\Sigma=\mu^{-1}(0)$ (a finite set) there exists a unique completion
$\tilde f:\tilde{\mathcal X}\rightarrow \tilde C$ of $\mu^*{\mathcal X}^0\rightarrow \tilde C\backslash\Sigma$ with $X_\infty=p^{-1}(\tilde 0)$ for all $\tilde 0\in \Sigma$.
\vskip .1in
Define
$$ Z^0\ = \ \{(t,z)\in C^0\times {\rm Hilb}({\mathbb P}^{N_m})\,:\ z\in {\mathcal T}_t\,\}
$$
where ${\mathcal T}_t$ is the set of all Hilbert points $[T(X_t)]$. Here $T:X_t\rightarrow{\mathbb P}^{N_m}$ ranges over the set of all Bergman imbeddings. In particular, ${\mathcal T}_t\subseteq {\rm Hilb}({\mathbb P}^{N_m})$ is a single  $G=SL(N_m+1)$  orbit.

\vskip .1in
We claim that $Z^0\subseteq C^0\times{\rm Hilb}({\mathbb P}^{N_m})$ is a constructible subset. To see
this, let $U\subseteq C^0$ be an affine open subset
and let $\sigma_0,...,\sigma_{N_m}$ be a fixed ${\mathcal O}(U)$ basis of ${\mathcal E}_m(U)$. This basis defines an imbedding

\beta\label{vb} S:\pi^{-1}(U)\ \rightarrow \ U\times{\mathbb P}^{N_m}
\ee
given by $x\mapsto (\pi(x), \sigma_0(x),...,\sigma_{N_m}(x))$. Define $H:U\rightarrow {\rm Hilb}({\mathbb P}^{N_m})$ by $H(t)= {\rm Hilb}(S(X_t))$ and define the map
$$ f_U:G\times U\ \rightarrow \ U\times{\rm Hilb}({\mathbb P}^{N_m})\ \
\hbox{given by $(g,t)\mapsto (t, g\cdot H(t))$}
$$
Then $f_U$ is an algebraic map so its image is constructible. This shows $Z^0|_{U}$ is
constructible for every affine subset $U\subseteq C^0$ and hence $Z^0$ is constructible.
\vskip .1in
Now we
fix $0<\epsilon<\epsilon(g)$ and
let $W_j=T_j(X_{t_j})$ where $T_j$ is the $\epsilon$-Bergman imbedding. Then (\ref{limit}) implies  $T_j(X_j)=W_j\rightarrow T_\infty(X_\infty)= W_\infty$, a stable algebraic curve in ${\mathbb P}^{N_m}$.
Let $Z\rightarrow C$ be the closure of $Z^0$ in $C\times {\rm Hilb}({\mathbb P}^{N_m})
\subseteq C\times{\mathbb P}^M$. Here ${\mathbb P}^M\supset {\rm Hilb}({\mathbb P}^{N_m})$ is chosen so that there is a $G$ action on ${\mathbb P}^M$ which restricts to the $G$ action on  ${\rm Hilb}({\mathbb P}^{N_m})$. Then $Z$ is a subvariety of $C\times {\rm Hilb}({\mathbb P}^{N_m})$ whose dimension we denote by $d$. Let $Z_t$ the fiber of $Z$ above $t\in C$.
 Then $[Y_\infty]\in Z_0$.
 \vskip .1in
To construct $\tilde C$ we use the Luna Slice Theorem:
There exists $W\subseteq{\mathbb C}^{M+1}$ a $G_{[Y_0]}$ invariant subspace such that
$[Y_\infty]\in {\mathbb P}(W)\subseteq{\mathbb P}^M$
and such that the map
$$ {\mathbb P}(W)\times {\rm Lie}(G)\ \rightarrow \ {\mathbb P}^{M}\ \ \hbox{
given by\ $(x,\xi)\mapsto \exp(\xi)x$}
$$
is a diffeomorphism of some small neighborhood $U_W\times V\subseteq {\mathbb P}(W)\times{\rm Lie}(G)$ onto an open set $\Omega\subseteq{\mathbb P}^M$,  with $U_W\subseteq{\mathbb P}(W)$ invariant under the finite group $G_{[Y_0]}$. After shrinking $U_W$ if necessary, the
intersection of a $G$ orbit with $U_W\backslash [Y_0]$ is a finite set of  order $m_1|m$ where
$m=|G_{[Y_0]}|$. In other words, the quotient $G_{[Y_0]}\backslash U_W$ parametrizes
the $G$-orbits in ${\mathbb P}^M$ that intersect $U_W$.
\vskip .1in
Note that $\Omega$ contains $(t_i, [Y_i])$ for infinitely many $i$ so $(C\times {\mathbb P}(W))\cap Z$ is a projective variety $C_1$ of dimension at least one. Moreover, if we let $C_2$ be the union of the components of $C_1$ containing $\{0\}\times [Y_\infty]$, then $C_2$ contains infinitely many of
$(t_i, [Y_i])$ so the image of $C_2\rightarrow C$ contains infinitely many $t_i$ and thus $C_2\rightarrow C$ is surjective. On the other hand, $C_2\rightarrow C$ is finite of degree $m_1$ (this follows from the construction of $U(W)$).
\vskip .1in
Let $\tilde C\subseteq C_1$ be an irreducible component of $C_1$ containing
$(t_i, [Y_i])$ for infinitely many $i$.
Let $H\subseteq G_{[Y_\infty]}$ be the set of all $\sigma\in
G_{[Y_\infty]}$ such that $\sigma(\tilde C)=\tilde C$. Then $H$ has order $d$ for
some $d|m_1$ and $\tilde C\rightarrow C$ is finite of degree $d$.
\vskip .1in
Finally, we have $\tilde C\subseteq Z\subseteq C\times {\rm Hilb}({\mathbb P}^{N_m})$. This gives us
a map $\tilde C\rightarrow {\rm Hilb}({\mathbb P}^{N_m})$. If we pull back the universal family
we get a flat family $\tilde {\mathcal X}\rightarrow \tilde C$ which extends
${\mathcal X}^0\times_{\tilde C}C^0$. This completes the proof of Theorem 1.2.\qed

\vskip .1in

\section{Uniqueness of the stable fill-in} \setcounter{equation}{0}
 Let $\pi:X^*\rightarrow\Delta^*\subset \Delta$ be an algebraic     family of smooth  curves genus $g$. We claim that
there exists a  unique stable analytic curve $X_0$ such that $X_t\rightarrow X_0$ in the Cheeger-Colding sense
as $t\rightarrow 0$.  This will establish the uniqueness statement of Theorem \ref{1point2}, and since existence was demonstrated in the previous section, it completes the proof.

\vskip .1in
Let $S: {\mathcal X}^*\rightarrow \Delta^*\times{\mathbb P}^{N_m}$ as in (\ref{vb}). For each $t\in\Delta^*$, the set $\underline\sigma_t=(\sigma_0(t),...,\sigma_{N_m}(t))$ is a basis of $H^0(X_t, mK_{X_t})$.
Let ${\underline s}_t=(s_0(t),...,s_{N_m}(t) )$ be the orthonormal basis
of $H^0(X_t, mK_{X_t})$ obtained by applying the Gram-Schmidt process
to  the basis $\underline\sigma_t$ and let $T^\epsilon_t: X_t\rightarrow{\mathbb P}^N$ be the map $T^\epsilon_t=T^\epsilon_{{\underline s}_t}$.
Here $0<\epsilon<\epsilon(g)$ is fixed once and for all. Remark \ref{consequence} implies that $t\mapsto [T_t^\epsilon(X_t)]$ defines a continuous function
$\Delta^*\rightarrow {\rm Hilb}$. Let

$$ z: \Delta^*\times SL(N+1,{\mathbb C})\ \rightarrow \ \Delta^*\times{\rm Hilb}
$$
and
$$f: \Delta^*\rightarrow \Delta^*\times {\rm Hilb}$$
be the maps
$$ z(t,g)\ = \ (t, g\cdot [T_t(X_t)])\ \
\hbox{and $f(t)\ = \ z(t, [T_t(X_t)])$.}
$$

\vskip .1in

Let $F=\overline{{\rm Im}(f)}\subseteq \Delta\times {\rm Hilb}$ and $Z=\overline{{\rm Im} z}\subseteq\Delta\times{\rm Hilb}$. Let $\pi_F:F\rightarrow \Delta$ and $\pi_Z:Z\rightarrow\Delta$
be the projection maps and $F_0=\pi_F^{-1}(0)$, $Z_0=\pi_Z^{-1}(0)$.
Observe that $F_0\subseteq{\rm Hilb}$ is closed and connected (this easily follows from the fact that $\Delta^*$ is connected and ${\rm Hilb}$ is compact and connected). Moreover, Theorem \ref{Tilimit} implies that every element of $F_0$ is of the form $T^\epsilon_{\underline s}(X_0)$ for some stable analytic curve $X_0$ and some basis ${\underline s}$.
\vskip .1in
\noindent
Claim:  $F_0$ is contained in the $U(N+1)$ orbit of $[X_0]$.
\vskip .1in
Assume the claim for the moment, and let's show that it implies uniqueness.
Suppose there exist subsequences $t_i, t_i'\in \Delta^*$ such that
$X_{t_i} \rightarrow X_0$ and $X_{t_i'}\rightarrow X_0'$. We must show that $X_0\approx X_0'$, i.e. $X_0$ and $X_0'$ are isomorphic stable analytic curves. Theorem \ref{Tilimit} implies there are bases ${\underline s}$ and ${\underline s}'$ such that $[T^\epsilon_{\underline s}(X_0)], [T^\epsilon_{{\underline s}'}(X_0')]\in F_0$ so
$ T^\epsilon_{{\underline s}'u}(X_0')\in U(N+1)\cdot T^\epsilon_{\underline s}(X_0)$. Now Lemma \ref {iso} implies  $X_0\approx X_0'$. This gives uniqueness.

\vskip .1in
\noindent
The set  $U=SL(N+1,{\mathbb C})\cdot [T^\epsilon_{\underline s}(X_0)]\subseteq Z_0$ is open since $\dim Z_0=\dim SL(N+1,{\mathbb C})$ and the stabilizer of $[T^\epsilon_{\underline s}(X_0]$ is finite.
Lemma \ref{iso} implies
\beta\label{U} F_0\cap U\ \subseteq \ U(N+1)[T_{{\underline s}}^\epsilon(X_0)] \ \subseteq \ U
\ee
Now $U(N+1)[T_{{\underline s}}^\epsilon(X_0)]$ is compact and $F_0$ is connected, so
$F_0\cap U = F_0$. Thus the claim follows from (\ref{U}). \qed
\vskip .1in

redpajama_set_name: RedPajamaArXiv


As a London based company with over a decade of experience, Muraldesign is commissioned by clients nationally and internationally to undertake bespoke projects of all sizes and styles. Every mural or canvas is expertly created to an exact brief and Muraldesign specialises in painting children's bedrooms, playrooms and nurseries along with trompe l'oeil and large scale murals.
Director Anna Starling (née Smith), the creative force behind the company, offers a complete service – from consultation through to completion – to ensure every commission exceeds expectations.


So, as most of you know Annabelle's birthday was last week and I feel like we are still recovering from the party. I got sick and Annabelle is actually still sick and Im hating every moment of it . The winter needs to go by quickly.
Anyway, now that Im getting back into the swing of things, I decided to start my youtube channel and what better topic to start with, other than my favorite instagram shops. When I was pregnant with Annabelle I started following Kid's fashion pages and from there I would see all these cute little babies with cute items on. Once I saw where the mommies would get their items from I started following a lot of these shops. I cant tell you how obsessed I am with all of them and just the overall talent that these women have is beyond amazing.
Below are some shops that were thoughtful enough to send Annabelle some cute items for her birthday and I want to tell you all how much we appreciation you guys thinking of us, we adore everything.
and please excuse the length, I promise all my videos wont be that long ,lol.
Hope you guys like this post and stay tuned for Annabelle's birthday post this Friday and some holiday DIY decor.


text:
using System;

namespace Problem5
{
    class Solution
    {
        private static int n = 3;
        private static int k = 2;
        private static int[] arr = new int[k];
        static string[] objects = {
            "hi", "a", "b"
        };

        // Write a recursive program for generating and printing all ordered k-element subsets from n-element set(variations Vkn).
        // Example: n=3, k=2, set =
        // {hi, a, b} → (hi hi), (hi a), (hi b), (a hi), (a a), (a b), (b hi), (b a), (b b)
        static void Main()
        {
            GenerateVariationsWithRepetitions(0);
        }

        static void GenerateVariationsWithRepetitions(int index)
        {
            if (index >= k)
            {
                PrintVariations();
            }
            else
            {
                for (int i = 0; i < n; i++)
                {
                    arr[index] = i;
                    GenerateVariationsWithRepetitions(index + 1);
                }
            }
        }

        static void PrintVariations()
        {
            Console.Write("(" + string.Join(", ", arr) + ") --> ( ");
            for (int i = 0; i < arr.Length; i++)
            {
                Console.Write(objects[arr[i]] + " ");
            }
            Console.WriteLine(")");
        }
    }
}

redpajama_set_name: RedPajamaGithub


text:
ELECTRO LEGEND DEAD
Florian Schneider dies aged 73 as tributes pour in for Kraftwerk founder
Jon Rogers
Kraftwerk founder Florian Schneider had died from cancer at the age of 73.
The German star founded the highly influential electronic group in 1970 with Ralf Hütter two years after meeting at the Academy of Arts in Remscheid.
Schneider was a founding member of the pioneering group Kraftwerk
Schneider appeared on the band's most influential albums, including Autobahn and Trans-Europe ExpressCredit: Getty Images - Getty
Schneider played the flute, violin and guitar, usually processed through electronic equipment.
He later said: "I found that the flute was too limiting.
"Soon I bought a microphone, then loudspeakers, then an echo, then a synthesiser. Much later I threw the flute away; it was a sort of process."
The band's breakthrough came with their fourth album Autobahn in 1974, which hit the US and UK charts.
This was followed by Radio-Activity, which saw the band invited to tour with David Bowie – an invite they declined.
Kraftwerk proceeded to release the iconic record Trans-Europe Express, followed by The Man-Machine and Computer World – the former featuring the single The Model, which hit number one in the UK and became their most commercial song.
They described their music as "industrielle volksmusik," which David Bowie translated as industrial folk music.
Afrika Bambaataa & the Soul Sonic Force sampled Kraftwerk for one of the earliest hip-hop hits, Planet Rock, while Computer World had a huge impact on the house and techno music that emerged from Chicago and Detroit that decade.
Bowie later paid homage to Schneider with his track V-2 Schneider on his "Heroes" album, which was partly recorded in Berlin while The Thin White Duke was living there.
Schenider remained part of the group up until 2008, when he didn't feature in the line-up when Kraftwerk played Coachella and a string of dates across the US and Europe, with video technician Stefan Pfaffe joining Hütter, Henning Schmitz and Fritz Hilpert.
Schneider said that he was working on different projects, and the following year, it was confirmed that he had left the band, with Pfaffe taking his place permanently.
Kraftwerk continued to tour, and planned to kick off their 50th anniversary tour in February, but this was postponed until 2021 due to the coronavirus outbreak.
In 2015, Schneider released a new piece of music, Stop Plastic Pollution, in collaboration with producer Dan Lacksman.
Tributes have been pouring in for the iconic musician.
Orchestral Manoeuvres in the Dark, said they were "absolutely devastated" and that he was one of their heroes.
Ultravox's Midge Ure said he was "way ahead of his time", while producer Thomas Dolby said: "Another of my great heroes gone."
Franz Ferdinand's Alex Kapranos remembered both Schneider and The Stranglers' Dave Greenfield, who also died this week, tweeting: "Sad to hear about Florian Schneider and Dave Greenfield. Two big inspirations in my life."
Schneider first started out on his musical career playing the fluteCredit: Getty - Contributor
Kraftwerk posed in Rotterdam, Netherlands in March 1976. L-R Florian Schneider, Karl Bartos, Wolfgang Flur and Ralf HutterCredit: Getty - Contributor
Kraftwerk performing live in Paris in 2002Credit: check copyright

redpajama_set_name: RedPajamaCommonCrawl


The Kingsway Entertainment District located on the Kingsway East in Sudbury, Canada will feature an Event Center/Arena, Festival Square, and a casino adjoining a hotel and associated restaurants. The Event Center will be owned by the City of Greater Sudbury while the gaming complex will be owned and operated by Gateway Casinos and Entertainment.
The Arena replacing the aging Sudbury Community Arena will have a seating capacity of 6,500 seats for concerts and 5,800 for hockey. The Festival Square connecting to the Casino/Hotel complex and Event Center will provide an outdoor entertainment space for family fun.
The project with an allocated budget of $100 million will be cost-shared between the City, Gateway Casinos and Entertainment, Zurich and company. Gateway has planned to invest $60 million and it will bring up to 250 new jobs thereby increasing its work force to over 400 employees in Greater Sudbury.
Construction works on site are expected to begin as soon as all necessary approvals are complete and the complex is expected to open in the spring of 2020.
There are currently no news regarding Kingsway Entertainment District.


27/06/2017 · OneNote becomes even more powerful at organizing your life when it's connected to the right add-ins and services. Save everything from meeting notes to recipes and even emails to your digital... I am trying to add additional notes to an invoice after I have received payment. For instance: Check number, last four of the Credit card used, etc.
The old algorithm was based on the pitch of the original note, which made comparisons difficult: we really don't know the pitch of the note we are trying to add, at least, not until we add it (since it depends on key, accidental state, etc). how to change the ip addres The old algorithm was based on the pitch of the original note, which made comparisons difficult: we really don't know the pitch of the note we are trying to add, at least, not until we add it (since it depends on key, accidental state, etc).
I am trying to add additional notes to an invoice after I have received payment. For instance: Check number, last four of the Credit card used, etc.
Once you've dragged the marker to the correct place, click the ADD DIRECTIONS HERE button. Enter the additional notes or directions on the form in the Directions/Notes field. You can also choose to display a label and/or symbol on the map if you wish, as in the example here.
A little note... You can also use Staff properties, (Right click staff and select a different [Instrument]), to change the pointer part way through a song. You can make your violin sound like a piano for three bars, then change it back.


We are pleased to announce our appointment as Central Agents for the stunning catamaran Dioni, a brand new Lagoon 42 of 2019.
This luxury catamaran offers a unique charter experience to her guests. She can comfortably accommodate up to 10 guests in 4 double en-suite cabins and two single cabins in the front peaks.
She is fully equipped with generator, air condition, solar panels.


"Don't Speed, Always Wear Your Seatbelt & Never Ever Drink & Drive"
Very happy to be hosting yet another Roger Rules Fundraiser at the world famous Blue Bird Cafe on Sept. 9th... I will release the line up soon... REMEMBER THE ROGER RULES: DON'T SPEED, ALWAYS WEAR YOUR SEAT BELT & NEVER EVER DRINK & DRIVE!!


Description: Join us for the second-annual Women in Compliance & Ethics event at the University of St. Thomas School of Law. Hear from established leaders in the field of compliance and ethics about the challenges and rewards of being a strong female leader in this space. We will focus our discussion on common and reoccurring trends in the marketplace, including the #metoo movement, Artificial Intelligence and how it may impact the ethics and compliance profession, and other relevant and timely topics.


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Manual senior wins national award for STEM training program for girls
Anjali Chadha won an award for Empowered, a nonprofit organization to educate and empower minority high school girls in the world of technology.
Manual senior wins national award for STEM training program for girls Anjali Chadha won an award for Empowered, a nonprofit organization to educate and empower minority high school girls in the world of technology. Check out this story on courier-journal.com: https://www.courier-journal.com/story/news/education/2018/10/05/prudential-spirit-community-award-empowered-stem-training/1497294002/
Elaine Rooker Jack, Special to Courier Journal Published 6:29 a.m. ET Oct. 5, 2018
duPont Manual student Anjali Chadha(Photo: Provided by Anjali Chadha)
MEET THE ACHIEVER
Achiever: Anjali Chadha
School: duPont Manual High School
Achievement: Anjali Chadha is one of two Kentucky winners of the Prudential Spirit of Community Awards for founding Empowered, a nonprofit organization that educates and empowers minority high school girls in the world of technology. The other Kentucky winner was Rachel Ritchie of Vine Grove. Anjali, 16, is a senior at duPont Manual High School.
Distinguished Finalists from Kentucky:
Hannah Huffaker, Glasgow
Colton Morrow, Monticello
Lydia O'Nan, Ashland
Christina Elizabeth Relich, Richmond
Background: The award was created in 1995 by Prudential in partnership with the National Association of Secondary School Principals. The program honors middle and high school students for outstanding service to others at the local, state and national levels, with the purpose to inspire others to contribute to their communities. To date, more than 125,000 young Americans have been recognized for their volunteer efforts.
Anjali's beginnings with technology: Anjali remembers being interested in computer programming and building websites by the age of 9, an interest she credits to her parents and "the world they lived in. They are smart and interesting people and their friends are smart and interesting people and involved in technology. That's what piqued my interest: the people seemed interesting."
You may like:How to watch Sundance-winning 'Science Fair' featuring Manual students
The roots of the idea for Empowered: Anjali attended Meyzeek Middle School, a math, science and technology magnet school. "It's also a reside school," said Anjali, "which means it's in a kind of rough part of Louisville, so under-privileged students attended with high achievers." In eighth grade she tutored in literacy, and she learned about her fellow students and their families. Many of their parents were small business owners, but they weren't using technology or social media to make those businesses successful. "I helped them build websites to help them build their businesses, but I realized I could be much more effective and make a bigger contribution if I trained other girls my age to do this work."
Empowered took shape: In the fall of her freshman year, Anjali attended an entrepreneurial experience now called Quarter Zero. "When I applied, I was the youngest person to go through it. I went to Boston once a month for three or four months and worked with business leaders — like from Google — and by the end of the program I'd made connections and had a working business model."
In the summer after her sophomore year she presented Empowered, a seven-week training program, for 10 girls. Anjali selected girls who met her criteria: girls 15 or older, with interest in business and technology, representing a diversity of Louisville schools, and whom she thought would most benefit from the program. Anjali knew that her pre-calculus teacher, Alisia McClain, was leaving Manual to get her Ph.D. She approached her about being operations manager for Empowered, and she agreed.
Participants would spend the majority of each day building websites and developing apps, using software they learned to use in the program. Anjali and McClain also planned team-building activities, and engaged high-profile women from the community for a speaker series. "It was inspirational, to see them making connections and meeting role models. These women were people who were rising above and making the world their own," Anjali said.
Achiever:Kentucky Country Day School junior scores perfect 36 on ACT exam
The payoff for Anjali: "I've become really involved in the Louisville entrepreneurial community. I've seen how it works and how it does great things. The payoff is also personal. I've learned a lot about what I need to learn and do better as a community leader."
She said in the first year of the program she was "detached, and I acted like I was an outside influence. I didn't interact with the girls, and I wished I'd done it differently. This year I changed my style and sincerely tried to learn about their interests. It was more enjoyable and enriching, and people appreciated it," she said. "It was a humbling experience, to see the power of drive, and the power of a mentor."
What's next for Empowered: Anjali will facilitate the program again this summer and is working to train others to keep it going, although she hasn't decided how that will work. "The entrepreneurial community has a great infrastructure but not a lot of kids," she said, citing the strength of the peer-to-peer connection. She said there are interested teachers in other parts of the state and she hopes to expand Empowered to other counties.
Her next goal is to plan "who and how to continue, and see how to grow it in scale."
For winning: Anjali received a silver medallion, an all-expense-paid trip to Washington, D.C., in May for recognition events — where 10 state winners were named National Honorees — and $1,000, which she intends to put back into Empowered. "You can never have enough money to run a program," she said, adding that she will plan more activities and field trips as well as apply for more grants.
Achiever:This North Oldham High School senior scored a perfect 36 on the ACT
Anjali's home life: She lives in the Springhurst area with her parents Vidya Ravichandran and Vikram Chadha and her younger sister Annika.
Teacher Quote: "Anjali is a phenomenal citizen," said McClain, "because she thinks outside of her community, outside of her age group, outside of her comfort zone. Because of this, she becomes an ally for others within our community, empowering everyone she meets. It is people like Anjali who change our world for the better."
Future plans: Anjali hopes to attend college in Boston. She said her passion is social entrepreneurship but she is also interested in hard science and research, so she is considering a biotechnology degree.
She enjoys: Classical dance, reading, writing and watching good movies.
Achiever:28 local students compete in Kentucky National Geographic State Bee
Read or Share this story: https://www.courier-journal.com/story/news/education/2018/10/05/prudential-spirit-community-award-empowered-stem-training/1497294002/
Report: McConnell's ancestors owned 14 slaves
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The FBI says gangs are using kids to carry out shooting spree


Most of us can attest to the fact that stress is reaching epidemic proportions in modern society. Balancing work, family, health, money, etc. is a challenge that many of us feel ill equipped to face. Eighty percent of the doctor visits in our country are stress related. Our quality of life and health is largely determined by how we adapt and relate to daily stressors. Excessive stress not only takes its toll on our bodies, but strips the joy out of life and suppresses our creative instincts. Without physical health, joy, and creativity, life is scarcely worth living. Dissolving stress is certainly possible, but takes a commitment to making lifestyle choices that create balance throughout our lives. Here are a few ancient techniques for eliminating stress, increasing energy, and creating emotional balance. These are some of the most powerful tools we have for achieving optimal health and preventing future disease.
1) Meditation: Practiced for thousands of years and in almost all cultures in various forms, meditation has long been recognized as one of the most powerful tools we have for cultivating peace of mind and balance. Numerous studies have proven the incredibly positive effect that meditation has on stress reduction. There are literally hundreds of meditation techniques taught around the world. For beginners, the most helpful approach is to start with basic mindfulness techniques that develop both relaxation and alertness. Once a basic ground of awareness has been stabilized, then more advanced meditation practices can be undertaken. Meditation is a practice that helps us identify with stillness and silence. It cultivates intuition and surrender. It can deeply help just about anybody, and is truly a miraculous practice for reducing stress and anxiety.
2) Yoga: This ancient practice has also been utilized by millions of people throughout history. Yoga is a form of meditation that involves putting the body into a variety of poses in combination with deep breathing to induce mental clarity, increased energy, and physical strength and flexibility. The healing benefits of yoga have been repeatedly documented by a variety of clinical studies. There are many forms of yoga and it is best to visit several teachers and studios to determine which form feels the most helpful for your needs. I highly recommend combining yoga with sitting meditation, as they work synergistically to induce deep states of relaxation.
3) Acupuncture: One of the pillars of traditional Chinese medicine, acupuncture has been practiced for at least 2,500 years. Acupuncture works by altering your blood flow to induce a variety of therapeutic effects. By bringing blood to an area which is injured or malfunctioning, it receives the oxygen, nutrients, hormones, and the bodies internally produced pain relievers to the afflicted area. The safety and efficacy of this practice are well documented, which accounts for its incredible surge in popularity in the Western world. Acupuncture's most potent 'side effect' is stress reduction. Be sure to see a licensed acupuncturist ( as opposed to chiropractors or MD's who practice acupuncture).
4) Herbal medicine: There are a variety of both Chinese and Western herbal formulas that have been clinically proven to be effective for many individual aliments, from 'simple' cold and flu and digestive complaints, to balancing hormones and women's cycles. Herbs are much less concentrated than pharmaceuticals, which is why they have far less side effects (but can still be as effective). If you are interested in this treatment option, many acupuncturists can skillfully prescribe herbal formulas. This is recommended over buying herbs over the counter with little knowledge of their intended usage.
5) Nutrition: Eating a diet high in antioxidants, essential fatty acids, and low glycemic carbohydrates can go a long way toward healing our 'modern diseases' of diabetes, high blood pressure, and heart disease. The standard American diet (high in processed foods, saturated fat, sugar, and transfats) has been linked to anxiety, depression, and increased stress in numerous studies. Change your diet to an organic, whole foods approach and both your body and mind will reward you beyond measure. Poor adaptability to stress is often a sign that our brains are starving for nutrients that we aren't getting from our standard American diets.
Making these lifestyle changes may not be easy in the initial phases. It is often helpful to seek out the support of a health care practitioner to guide you through these transitions. Once you start feeling the enormous payoff of making such changes, there truly is no turning back. Your stress will dissolve, your weight will decrease, and your energy will skyrocket, not to mention the preventative measures you are taking for heart health. Isn't that enough to warrant making a few changes?


A LEADING MANUFACTURER OF LABELS, STICKERS, NAMEPLATES AND OTHER BRANDING PRODUCTS.
Stickers and labels occupy a distinctive place when it comes to branding of a product. They have the potential to increase the worth of the product by many folds in the eyes of the buyer. Most of the time it must meet extremely critical test procedures so as to be accepted. Shivlik Enterprises is a leading manufacturer and exporter of Labels, Name plates, Logos and many more related products as mentioned below. We are confident of meeting this quality requirement. Our core strength lies in our Quality, Expertise and our wide range of products.


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Douglas McGrath(1958-2022)
Company Man (2000)
Douglas McGrath was born on 12 February 1958 in Midland, Texas, USA. He was a writer and actor, known for Emma (1996), Bullets Over Broadway (1994) and Small Time Crooks (2000). He was previously married to Jane Read Martin. He died on 3 November 2022 in Manhattan, New York City, New York, USA.
DiedNovember 3, 2022(64)
1 win & 10 nominations total
Bullets Over Broadway
Small Time Crooks
Frenchy's Lawyer
Tom(as Doug McGrath)
screenplay: musical book
The Steven Banks Show
Rifkin's Festival
Gil Brenner
Edward Solomon
Crisis in Six Scenes
Principal Toby Cook
Solitary Man
Dean Edward Gitleson (as Doug McGrath)
Michael Clayton
Jeff Gaffney (as Doug McGrath)
Hollywood Ending
Barbeque Guest
Alan Quimp
Private Investigator (as Doug McGrath)
Bill Gaines
Tom (as Doug McGrath)
Prix Fixe
Bob Waterman
Becoming Mike Nichols
I Don't Know How She Does It
Director (as Doug McGrath)
His Way
France's national library catalogue
Theater_org
Doug McGrath
Midland, Texas, USA
Manhattan, New York City, New York, USA(heart attack)
Jane Read MartinJune 3, 1995 - November 3, 2022 (his death, 1 child)
Former columnist for The New Republic.
He has appeared in seven works of Woody Allen: Celebrity (1998), Sweet and Lowdown (1999), Small Time Crooks (2000), Hollywood Ending (2002), Café Society (2016), in the first episode of Crisis in six scenes (2016), and Rifkin's Festival (2020).


Order the newest Livelyhood show!
In an increasingly round-the-clock economy, we can have online groceries delivered before breakfast, get a tan at midnight, and drop kids off at a 24-hour daycare. Who works the dawn to dusk jobs that make all this possible? Livelyhood finds out.
LIVELYHOOD is more than just great television!
This eight part series of PBS specials are designed to encourage a dialogue in classrooms, boardrooms, lunchrooms, and libraries about how companies, communities, and co-workers can find new ways to cope with the rapidly changing workplace. These videos are brought to you by The Working Group.
This off-beat and often uplifting series of specials looks with uncommon humor and grace at a wide-ranging and vivid group of real people.
PBS' Livelyhood series is full of hope. One of the most inspiring hours of television this year.
I teach high school Home Economics. My program includes career preparation, and in Texas we have a single semester called Family and Career Management. The Working Family Values show is perfect. I wish I had a new class starting tomorrow.
I caught a couple of the shows--on the work ethic and on employee ownership--and thought they were great. Interesting, entertaining, informative, and provocative, I decided to show them to my classes at UC Davis. They loved them too! Thanks and keep up the good work.
AWESOME Show! ... great production, stories, etc. Just a really unexpected treasure to find on TV. The educational value alone is tremendous. Keep up the great work.
Founded in 1988, The Working Group (TWG) is a media production company that has built its reputation by turning its cameras toward a group rarely seen on television: ordinary, hard-working Americans.
Designed to get people talking to each other, TWG videos are educational, entertaining, and great resources for discussions in classrooms, conferences, and workplaces.
The Working Group (TWG) is a 501(c)(3) non-profit organization incorporated in the state of California.
Major funding has been provided by the W.K. Kellogg Foundation, the UAW-DaimlerChrysler National Training Center and United Airlines. Livelyhood is presented by KQED-TV in San Francisco.