text:
\section{Introduction}

A classical theorem, due to the combined work of Almgren, Pitts and Schoen--Simon,
asserts that for $n \geq 2$, every $(n+1)$-dimensional closed Riemannian manifold $M$ contains a
minimal hypersurface smoothly embedded away from a closed singular set of Hausdorff
dimension at most $n-7$. The original proof of this theorem is based on a highly
non-trivial geometric min-max construction due to Pitts~\cite{Pitts81},
which extended earlier work of Almgren~\cite{Almgren65}.
This construction is carried out directly for
the area functional on the space of hypersurfaces equipped with an appropriate weak
topology, and it yields in the first instance a critical point of area satisfying
a certain almost-minimizing property.
This property is central to the rest of the argument, and allows
to deduce regularity of the min-max hypersurface from compactness of
the space of uniformly area-bounded stable minimal hypersurfaces
with singular sets of dimension at most $n-7$,
a result proved for $2 \leq n \leq 5$ by Schoen--Simon--Yau~\cite{SchoenSimonYau75} and
extended to arbitrary $n \geq 2$ by Schoen--Simon~\cite{SchoenSimon81}.
(The Almgren--Pitts min-max construction has recently been streamlined by
De Lellis and Tasnady~\cite{DeLellisTasnady13} giving a shorter proof.
However, their argument still follows Pitts' closely and is in
particular based on carrying out the min-max procedure directly
for the area functional on hypersurfaces.)

In recent years an alternative approach to this theorem has been developed, whose philosophy is to push the regularity theory to its limit in order to gain substantial simplicity on the existence part.
Specifically, this approach differs from the original one in two key aspects:
first, it is based on a strictly PDE-theoretic min-max construction that replaces the Almgren--Pitts geometric construction;
second, for the regularity conclusions, it relies on a sharpening of
the Schoen--Simon compactness theory for stable minimal hypersurfaces.
The idea in this approach is to construct a minimal hypersurface as the
limit-interface associated with a sequence of solutions $u = u_{i}$ to the Allen--Cahn equation
\begin{equation}
\label{eq:allen_cahn_equation}
\Delta u -\epsilon_{i}^{-2} W'(u) = 0
\end{equation}
on the ambient space $M$, where $W \! : {\mathbb R} \to {\mathbb R}$
is a fixed double-well potential with precisely two minima at $\pm 1$ with $W(\pm1) = 0$.
Roughly speaking, if the $u_{i}$ solve \eqref{eq:allen_cahn_equation} and satisfy appropriate bounds, then the level sets of $u_i$
converge as $\epsilon_{i} \to 0^{+}$ to a stationary codimension
$1$ integral varifold $V$.
This fact was rigorously established by
Hutchinson--Tonegawa~\cite{HutchinsonTonegawa00}, using in part methods inspired
by the earlier work of Ilmanen in the parabolic setting Ilmanen~\cite{Ilmanen93}.
Note that $u_{i}$ solves \eqref{eq:allen_cahn_equation} if and only if
it is a critical point of the Allen--Cahn functional
\begin{equation}
E_{\epsilon_{i}} (u)
= \int_U \epsilon_{i} \frac{\abs{\nabla u}^{2}}{2} + \frac{W(u_i)}{\eps_i}.
\end{equation}
If the solutions $u_{i}$ are additionally assumed stable with respect to
$E_{\epsilon_{i}}$, then Tonegawa and Wickramasekera \cite{TonegawaWickramasekera10}
proved that the resulting
varifold $V$ is supported on a hypersurface smoothly embedded away from a closed
singular set of Hausdorff dimension at most $n-7$, using an earlier result of
Tonegawa \cite{Tonegawa05} which established the stability of the regular part
$\reg V$ with respect to the area functional.
Their proof of this regularity result uses the regularity and compactness theory
for stable codimension 1 integral varifolds developed by
Wickramasekera~\cite{Wickramasekera14} sharpening the Schoen--Simon theory.

Stability of $u_i$ means that the second variation of the Allen--Cahn functional
$E_{\eps_i}$ with respect to $H^1(M)$ is a non-negative quadratic form.
More generally the index $\ind u_i$ denotes the number of strictly negative
eigenvalues of the elliptic operator $L_i = \Delta - \eps_i^{-2}W''(u_i)$,
so that $u_i$ is stable if and only if $\ind u_i = 0$.
Using min-max methods for semi-linear equations,
Guaraco~\cite{Guaraco15} recently gave a simple and elegant construction
of a solution $u_{i}$ to \eqref{eq:allen_cahn_equation}
with $\ind  u_{i} \leq 1$ and $\norm{u_i}_{L^\infty} \leq~1$,
and such that as $\epsilon_{i} \to 0$, the energies
$E_{\epsilon_{i}}(u_{i})$ are bounded above and below away from 0.
The lower energy bound
guarantees that the resulting limit varifold $V$ is non-trivial.
Since $\ind u_i \leq 1$, $u_i$ must be stable in at least one of
every pair of disjoint open subsets of $M$; similarly if
$\ind u_i \leq k$ then $u_i$ must be stable in at least one of every
$(k+1)$-tuple of disjoint open sets. This elementary observation,
originally due to Pitts in the context of minimal surfaces, together
with a tangent cone analysis in low dimensions, allowed Guaraco
to deduce the regularity of $V$ from the results of \cite{TonegawaWickramasekera10}.
More recently still, Gaspar and Guaraco~\cite{GasparGuaraco2016}
have used $k$-parameter min-max methods to produce sequences of
critical points with Morse index at most $k$, for all positive
integers~$k$.
Our results show that this index bound is inherited by
the minimal surface arising as $\eps_i \to 0$, provided
it has a trivial normal bundle.
We also point out that the regularity follows
in all dimensions from the corresponding result in the stable case via
an inductive argument that avoids the tangent cone analysis used in~\cite{Guaraco15}.

\begin{cor*}
Let $M$ be a closed Riemannian manifold of dimension $n+1 \geq 3$.
Let $V$ be the integral varifold arising as the limit-interface of
the sequence $(u_i)$ of solutions to \eqref{eq:allen_cahn_equation}
constructed in~\textup{\cite{Guaraco15}}
(respectively in~\textup{\cite{GasparGuaraco2016}} using $k$-parameter min-max	methods).
Then $\dimh \sing V \leq n-7$. If $\reg V$ is two-sided,
then its Morse index with respect to the
area functional satisfies $\ind_{\calH^n} \reg V \leq 1$
(respectively $\ind_{\calH^n} \reg V \leq k$).
\end{cor*}
In min-max theory, one generally expects that the Morse index of the constructed critical point is no greater than the number of parameters
used in the construction.
The above corollary gives this result for the constructions of Guaraco and
Gaspar--Guaraco, provided the arising hypersurface is two-sided.
This was recently shown by Chodosh and Mantoulidis~\cite{ChodoshMantoulidis2018}
to hold automatically when the ambient manifold $M$
has dimension $3$ and is equipped either with a bumpy metric or has positive
Ricci curvature. Building on work of Wang and Wei~\cite{WangWei2017}, Chodosh--Mantoulidis
prove curvature and strong sheet separation estimates, and use these
to deduce that in this three-dimensional setting the convergence of the level
sets occurs with multiplicity $1$.
They moreover show that in all dimensions, if the limiting surface has
multiplicity $1$, then its index is bounded \emph{below} by the index
of the $u_\eps$.

This complements our upper bound for the index, which is a direct consequence
of a lower bound for $(\lambda_p)$, the spectrum of the elliptic
operator $L_V = \Delta_V + \abs{A}^2 + \Ric_M(\nu,\nu)$---the \emph{scalar
Jacobi operator}---in terms of $(\lambda_p^i)$, the spectra of the operators $(L_i)$.
Establishing this spectral lower bound is our main result.
\begin{thm*}
Let $M$ be a closed Riemannian manifold of dimension $n+1 \geq 3$.
Let $V$ be the integral
varifold arising from a sequence $(u_i)$ of solutions to
\eqref{eq:allen_cahn_equation} with $\ind u_i \leq k$ for some $k \in \NN$.
Then $\dimh \sing V \leq n-7$ and
\begin{enumerate}[font = \upshape, label = (\roman*)]
	\item $\lambda_p(W) \geq \limsup_{i \to \infty} \lambda_p^i(W)$ for all
	$W \subset \subset M \setminus \sing V$ and $p \in \NN$, \label{item:intro_thm_spec_lower_bd}
	\item $\ind_{\calH^n} C \leq k$ for every two-sided connected component
	$C \subset \reg V$.
\end{enumerate}
\end{thm*}
\begin{rem*}
The spectral lower bound of~\ref{item:intro_thm_spec_lower_bd} also holds
if the assumptions on the $u_i$ are weakened in a spirit similar to the	work of
Ambrozio, Carlotto and Sharp~\cite{AmbrozioCarlottoSharp2015},
that is if instead of an index upper bound one assumes that for some
$k \in \NN$ there is $\mu \in \RR$ such that $\lambda_k^i \geq \mu$ for all $i$.
(Note that the index bound $\ind u_i \leq k$ is equivalent to
$\lambda_{k+1}^i \geq 0$.)
\end{rem*}
\begin{rem*}
It was recently brought to our attention that a similar result
had previously been proved by Le~\cite{Le2011} in ambient Euclidean space,
under the additional assumption that the convergence to the
limit surface occurs with multiplicity $1$.
Adapting the methods developed in~\cite{Le2011,Le2015}
to ambient Riemannian manifolds, Gaspar generalised our
results to the case where the limit varifold is one-sided,
without any assumption on multiplicity~\cite{Gaspar2017}.
Their general approach is similar to ours but subtly different,
in that they instead consider the second \emph{inner}
variation of the Allen--Cahn functional; see also the recent
work of Le and Sternberg~\cite{LeSternberg2018}, where similar bounds are established
for other examples of eigenvalue problems.
\end{rem*}

For the minimal hypersurfaces obtained by a direct min-max
procedure for the area functional on the space of hypersurfaces (as in the Almgren--Pitts construction), index bounds have recently been established
by Marques and Neves~\cite{MarquesNeves15}.
Both the Almgren--Pitts existence proof and the Marques--Neves proof of
the index bounds are rather technically involved; in particular, the
min-max construction in this setting has to be carried out in a bare-handed
fashion in the absence of anything like a Hilbert space structure.
In contrast, in the approach via the Allen--Cahn functional, Guaraco's
existence proof is strikingly simple, and our proofs for the spectral
bound and the regularity of $V$ are elementary bar the fact that
they rely on the highly non-trivial sharpening of the
Schoen--Simon regularity theory for stable hypersurfaces as
in~\cite{Wickramasekera14}.

\begin{outline}
In Section~\ref{sec:main_statement} we briefly expose notions from the theory of
varifolds, set the context for the rest of the paper and give the statements
of the main result and its corollaries.
Their proof requires a number of preliminary results, which are contained in
Section~\ref{section:preliminary_results}.
The proof of the main result (Theorem~\ref{thm:main_thm_propagation_of_index_bounds})
is in Section~\ref{sec:proof_main_thm}, and is split into two parts:
in the first part we prove the spectral lower bound by an inductive
argument on $\ind u_i$; this immediately implies the
index upper bound. The proof of $\dimh \sing V \leq n-7$ is given in the second
part, and uses a similar inductive argument.
There are two appendices: Appendix~\ref{app:measure_function_convergence} contains
two elementary lemmas from measure theory that are used repeatedly in
Section~\ref{section:preliminary_results}.
Appendix~\ref{app:second_fundamental_form_coordinate_expression}
gives a proof of Proposition~\ref{prop:weak_convergence_second_ff},
which is a straight-forward adaptation of an argument used by
Tonegawa for the stable case.
\end{outline}

\begin{ackn}
I would like to thank my PhD supervisor Neshan Wickramasekera for his
encouragement and support, and Otis Chodosh for helpful conversations.
This work was supported by the UK Engineering and Physical Sciences Research Council
(EPSRC) grant EP/L016516/1 for the University of Cambridge Centre for Doctoral
Training, the Cambridge Centre for Analysis.
\end{ackn}

\section{Varifolds, stability and statement of main theorem}
\label{sec:main_statement}
The setting is as follows:
$(M^{n+1},g)$ is a closed (that is, compact without boundary) Riemannian manifold
of dimension $n+1 \geq 3$, and $U \subset M$ is an arbitrary
open subset, possibly equal to $M$ itself.

\subsection{Varifolds: basic definitions}
An $n$-dimensional \emph{varifold} in $U$ is a Radon measure
on the Grassmannian $G_n(U) = \{(p,E) \mid p \in U, E \subset T_pM,
\dim E = n\}$---the space of $n$-dimensional planes over points in $U$.

An important subclass are the \emph{integral varifolds},
which correspond to a pair $(\Sigma,\theta)$ of a countably $n$-rectifiable set
$\Sigma \subset U$ and a Borel-measurable function $\theta \in L_{\mathrm{loc}}^1(\Sigma,\NN)$ via
\begin{equation}
V_{\Sigma,\theta}(\phi) = \int_U \phi(x,T_x\Sigma) \theta(x) \dH^n(x)
\quad \text{for all $\phi \in C_c(G_n(U))$},
\end{equation}
where $T_x\Sigma$ is the $\calH^n$-a.e. defined tangent space to $\Sigma$.
The function $\theta$ is called the \emph{multiplicity function}.

A sequence $(\vi)$ \emph{converges as varifolds} to $V$
if they converge weakly as Radon measures on $G_n(U)$, i.e.\ if
\begin{equation}
\int_{G_n(U)} \phi \dvi \to \int_{G_n(U)} \phi \dv
\quad
\text{for all $\phi \in C_c(G_n(U))$.}
\end{equation}

The \emph{weight measure} $\norm{V}$ of a varifold $V$ is defined by
\begin{equation}
	\norm{V}(\phi) = \int_{G_n(U)} \phi(x) \dv(x,S)
\quad
\text{for all $\phi \in C_c(U)$}.
\end{equation}

Consider an arbitrary vector field $X \in C_c^1(U,TM)$ with
flow $(\Phi_t)$. We deform $V$ in the direction
of $X$ by pushing it forward via its flow, that is
\begin{equation}
(\Phi_t)_*V(\phi) = \int_{G_n(U)} \phi(\Phi_t(x),\intdiff \Phi_t(x) \cdot S) J \Phi_t(x) \dv(x,S)
\end{equation}
for all $\phi \in C_c(G_n(U))$,
where $J\Phi_t(x) =
\det (\mathrm{d} \Phi_t(x)^* \circ \mathrm{d} \Phi_t(x)) ^{\frac{1}{2}}$
is the Jacobian of $\Phi_t(x)$.

Differentiating the corresponding weight measures
$\lVert (\Phi_t)_* V \rVert$ yields the \emph{first variation} of $V$:
\begin{equation}
\delta V(X) = \srestr{\frac{\mathrm{d}}{\mathrm{d}t}}{t = 0} \norm{(\Phi_t)_*V}(U).
\end{equation}
When $\delta V(X) = 0$ for all vector fields $X \in C_c^1(U,TM)$, we say
that $V$ is \emph{stationary} in $U$.

By definition, the \emph{regular part} of $V$ is the set of points
$x \in U \cap \spt \lVert V \rVert$ such that in a neighbourhood of $x$,
$\spt \lVert V \rVert$ is smoothly embedded in $M$.
Its complement is the \emph{singular part} of $V$,
denoted $\sing V := U \cap \spt \nv \setminus \reg V$.
For a stationary integral varifold $V$, the Allard regularity theorem
implies that $\reg V$ is a dense subset of $U \cap \spt \lVert V \rVert$
\cite[Ch.~5]{Simon84}.
\subsection{Stability and the scalar Jacobi operator}
Throughout this section $V$ will be a stationary integral $n$-varifold in $U \subset M$.
We call $V$ \emph{two-sided} if its regular part $\reg V$
is two-sided, that is if the normal bundle $NV := N(\reg V)$
admits a continuous non-vanishing section.
When this fails, $V$ is called \emph{one-sided}.
(Recall that when the ambient manifold $M$ is orientable,
then $\reg V$ is two-sided if and only if it is orientable.)

Suppose that $V$ is two-sided, and fix a unit normal vector field
$N \in C^1(NV)$, so that every function $\phi \in C_c^1(\reg V)$
corresponds to a section $\phi N \in C_c^1(NV)$ and vice-versa.
After extending the vector field $\phi N$ to $C_c^1(U,TM)$---the chosen
extension will not matter for our purposes---we can deform $\reg V$
with respect to its flow $(\Phi_t)$.
As $V$ is stationary, the first variation vanishes: $\delta V(\phi N) = 0$.
A routine calculation, the details of which can be found for instance
in~\cite[Ch.~2]{Simon84} shows that the second variation satisfies
\begin{multline}
\label{eq:second_variation_smooth}
\delta^2 V(\phi N) =
\srestr{\frac{\intdiff^2}{\intdiff t^2}}{t = 0} \norm{(\Phi_t)_* V}(U)
= \\ \int_U \abs{\nabla_V \phi}^2 -
( \abs{A}^2 + \Ric_M(N,N) )\phi^2 \dnv,
\end{multline}
where $\nabla_V$ is the Levi-Civita connection on $\reg V$,
$A$ is the second fundamental form of $\reg V \subset M$, and
$\Ric_M$ is the Ricci curvature tensor on $M$.

The expression on the right-hand side can be defined for one-sided
$V$ by replacing $N$ by an arbitrary measurable unit section
$\nu: \reg V \to NV$, but it loses its interpretation in terms
of the second variation of the area.

\begin{defn}[Scalar second variation]
The \emph{scalar second variation} of a stationary integral varifold $V$
is the quadratic form $B_V$ defined for $\phi \in C_c^2(\reg V)$ by
\begin{equation}
\label{defn:scalar_second_variation}
B_V(\phi,\phi)
= \int_{\reg V} \abs{\nabla_V \phi}^2
- (\abs{A}^2 + \Ric_M(\nu,\nu)) \phi^2 \dnv.
\end{equation}

\end{defn}
\begin{rem}
When $V$ is one-sided, the second variation of its area has to be measured
with respect to variations in $C_c^1(NV)$---we refer
to \cite[Ch.~2]{Simon84} or~\cite[Sec.~1.8]{ColdingMinicozzi11}
for further information on this.
We called $B_V$ `scalar' in order to highlight its difference with
the second variation of area in this case,
but emphasise that for the remainder `second variation' refers exclusively to the
quadratic form $B_V$ from Definition~\ref{defn:scalar_second_variation}.
(For the same reasons we also call the Jacobi operator $L_V$
`scalar' in Definition~\ref{defn:scalar_jac} below,
but omit this adjective in the remainder of the text.)
\end{rem}
One can consider $\reg V$ as a stationary integral varifold
in its own right by identifying it with the corresponding varifold
with constant multiplicity $1$. Its scalar second variation
\begin{equation}
B_{\reg V}(\phi,\phi) = \int_{\reg V} \abs{\nabla_V \phi}^2
- (\abs{A}^2 + \Ric_M(\nu,\nu)) \phi^2 \dH^n
\end{equation}
differs from $B_V$ only in that the integral is with respect
to the $n$-dimensional Hausdorff measure instead of $\nv$. This means exactly that
while $B_V$ is `weighted' by the multiplicity of $V$, the quadratic
form $B_{\reg V}$ measures the variation of `unweighted' area;
we will briefly use this in Section~\ref{sec:prelim_spectrum_sec_var}.

After integrating by parts on $\reg V$, the form $B_V$
corresponds to the second-order elliptic operator
$L_V = \Delta_V + \abs{A}^2 + \Ric_M(\nu,\nu),$
where $\Delta_V$ is the Laplacian on $\reg V$.
\begin{defn}[Scalar Jacobi operator]
	\label{defn:scalar_jac}
	The \emph{scalar Jacobi operator} of $V$, denoted $L_V$, is the second-order
	elliptic operator
	\begin{equation}
	L_V \phi = \Delta_V \phi +
	(\abs{A}^2 + \Ric_M(\nu,\nu)) \phi
	\quad \text{for all $\phi \in C^2(\reg V)$},
	\end{equation}
	where $\nu: \reg V \to NV$ is an arbitrary measurable unit normal
	vector field.

\end{defn}
The curvature of $\reg V$ can blow up as one approaches $\sing V$,
in which case the coefficients of the operator $L_V$ would
not be bounded. To avoid this, we restrict ourselves to a compactly
contained open subset $W \subset \subset U \setminus \sing V$;
moreover we require
$W \cap \reg V \neq \emptyset$ to avoid vacuous statements.

We use the sign convention for the spectrum defined in
\cite[Ch.~8]{GilbargTrudinger98}, where $\lambda \in \RR$
is an eigenvalue of $L_V$ in $W$ if there is $\varphi
\in H_0^1(W \cap \reg V)$ such that $L_V \varphi + \lambda \varphi = 0$.
By standard elliptic PDE theory the spectrum
\begin{equation}
\lambda_1 \leq \lambda_2 \leq \cdots \to +\infty
\end{equation}
of $L_V$ in $W$ is discrete and bounded below.
We will sometimes also write $\lambda_p(W)$ instead of $\lambda_p$
in order to highlight the dependence of the spectrum on the
subset $W$.
The eigenvectors of $L_V$
span the space $H_0^1(W \cap \reg V) = W_0^{1,2}(W \cap \reg V)$,
which we abbreviate throughout by $H_0^1$.

The \emph{index of $B_V$} in $W$ is the
maximal dimension of a subspace of $H_0^1$ on which
$B_V$ is negative definite; equivalently
\begin{equation}
\ind_W B_V = \mathrm{card} \{ p \in \NN \mid \lambda_p(W) < 0 \}.
\end{equation}
Moreover $\ind B_V := \sup_W (\ind_W B_V)$, where
the supremum is taken over all $W \subset \subset U \setminus \sing V$ with
$W \cap \reg V \neq \emptyset$.
\begin{rem}
	We will see in Section~\ref{sec:prelim_spectrum_sec_var}
	that the index of $B_V$ coincides with the Morse index
	of $\reg V$ with respect to the area functional,
	at least when $\reg V$ is two-sided.
\end{rem}

\subsection{Statement of main theorem}

Let $(\eps_i)$ be a sequence of positive parameters with $\eps_i \to 0$
and consider an associated sequence of functions $(u_i)$ in $C^3(U)$ satisfying the
following hypotheses:

\begin{enumerate}[label=(\Alph*)]
\item Every $u_i \in C^3(U)$ is a critical point of the Allen--Cahn functional
\begin{equation}
\label{eq:allen_cahn_functional}
E_{\epsilon_i}(u)
= \int_U
\epsilon_i \frac{\lvert \nabla u \rvert^2}{2} + \frac{W(u)}{\epsilon_i}
\dH^{n+1},
\end{equation}
i.e. $u_i$ satisfies the equation
\begin{equation}
\label{eq:epsilon_allen_cahn_equation}
- \epsilon_i^2 \Delta u_i + W'(u_i) = 0 \quad \text{in } U.
\end{equation} \label{hypA}

\item There exist constants $C, E_0 < \infty$ such that
\begin{equation}
\sup_i \lVert u_i \rVert_{L^\infty(U)} \leq C\; \text{and} \; \sup_i E_{\epsilon_i}(u_i) \leq E_0.
\end{equation} \label{hypB}

\item There exists an integer $k \geq 0$ such that the Morse index of each $u_i$
is at most $k$, i.e. any subspace of $C_c^1(U)$ on which the second variation
\begin{equation}
\delta^2 E_{\epsilon_i}(u_i)(\phi,\phi)
= \int_U \epsilon_i \lvert \nabla \phi \rvert^2
+ \frac{W''(u_i)}{\epsilon_i} \phi^2
\dH^{n+1}
\end{equation}
is negative definite has dimension at most $k$. We write this $\ind u_i \leq k$,
and if $k = 0$, say that $u_i$ is \emph{stable in $U$}.
\label{hyp:C_index_bound}
\end{enumerate}
\begin{rem}
\label{rem:defn_stability}
More generally  $\ind_{U'} u_i$ denotes
the index of $\delta^2 E_{\eps_i}(u_i)$ with respect to variations in
$C_c^1(U')$ (or equivalently in $H_0^1(U')$) for all open
subsets $U' \subset U$.
When $\ind_{U'} u_i = 0$, we say that $u_i$ is \emph{stable in U'}.
\end{rem}

We follow Tonegawa~\cite{Tonegawa05}, using an idea originally
developed by Ilmanen~\cite{Ilmanen93} in a parabolic setting
to `average the level sets' of $u_i \in C^3(U)$
and define a varifold $\vei$ by
\begin{equation}
\label{eq:definition_varifolds}
\vei(\phi)
= \frac{1}{\sigma}
\int_{U \cap \{\nabla u_i \neq 0\}}
\epsilon_i \frac{\lvert \nabla u_i(x) \rvert^2}{2}
\phi(x,T_x \{u_i = u_i(x)\})
\dH^{n+1}(x)
\end{equation}
for all $\phi \in C_c(G_n(U))$.
Here $T_x \{u_i = u_i(x)\}$ is the tangent space to the level set
$\{u_i = u_i(x)\}$ at $x \in U$,
and $\sigma = \int_{-1}^1 \sqrt{W(s)/2} \intdiff s$ is a constant.
\begin{rem}
In~\cite{HutchinsonTonegawa00,Guaraco15} the varifold $\vei$ is defined by
the slightly different expression
$	\vei(\phi)
= \frac{1}{\sigma}
\int_{U \cap \{\nabla u_i \neq 0\}}
\abs{\nabla w_i(x)}  \phi(x,T_x \{u_i = u_i(x)\}) \dH^{n+1}(x)$, with
$w_i$ as in Theorem~\ref{lem:properties_V}.
The `equipartition of energy' \eqref{eq:thm_properties_V_equipartition_energy}
from Theorem~\ref{lem:properties_V}
shows that the two definitions give rise to the same limit varifold $V$
as $i \to \infty$.
\end{rem}
The weight measures $\norm{\vei}$ of these varifolds satisfy
\begin{equation}
\label{eq:density_weight_measure}
\lVert \vei \rVert(A)
= \frac{1}{\sigma} \int_{A \cap \{\nabla u_i \neq 0\}}
\epsilon_i \frac{\lvert \nabla u_i \rvert^2}{2} \dH^{n+1}
\leq \frac{E_0}{2\sigma}
\end{equation}
for all Borel subsets $A \subset U$, where the inequality follows
from the energy bound in Hypothesis~\ref{hypB}.
The resulting bound $\vei(G_n(U)) \leq \frac{E_0}{2\sigma}$ allows us to extract a
subsequence that converges to a varifold $V$, with properties laid out in the following theorem by Hutchinson--Tonegawa~\cite{HutchinsonTonegawa00}.
\begin{thm}[\cite{HutchinsonTonegawa00}]
	\label{lem:properties_V}
	Let $(u_i)$ be a sequence in $C^3(U)$ satisfying Hypotheses (A) and (B).
	Passing to a subsequence $\vei \rightharpoonup V$ as varifolds, and
	\begin{enumerate}[font = \upshape, label = (\alph*)]
		\item $V$ is a stationary integral varifold,
		\item $\nv(U) = \liminf_{i \to \infty} \frac{1}{2 \sigma} E_{\eps_i}(u_i),$
		\item for all $\phi \in C_c(U)$:
		\begin{equation}
		\label{eq:thm_properties_V_equipartition_energy}
		\lim_{i \to \infty}	\int_U
		\epsilon_i \frac{\abs{\nabla u_i}^2}{2} \phi
		= 	\lim_{i \to \infty} \int_U \frac{W(u_i)}{\epsilon_i} \phi
		= 	\lim_{i \to \infty} \int_U \abs{\nabla w_i} \phi,
		\end{equation}
		where $w_i := \Psi \circ u_i$ and
		$\Psi(t) := \int_{0}^{t} \sqrt{W(s)/2} \intdiff s$.

	\end{enumerate}
\end{thm}

Up to a factor of $\eps_i$ the second variation $\delta^2 E_{\eps_i}$
corresponds to the second-order elliptic operator
$L_i := \Delta - \eps_i^{-2} W''(u_i)$. As in the discussion for
the Jacobi operator, $L_i$
has discrete spectrum $\lambda_1^i \leq \lambda_2^i \leq \cdots \to +\infty$,
which we denote by $\lambda_p^i(W)$ when we want to
emphasise its dependence on the subset $W$.
The following theorem is our main result.
\begin{customthm}{A}
\label{thm:main_thm_propagation_of_index_bounds}
Let $M^{n+1}$ be a closed Riemannian manifold, and $U \subset M$ an open subset.
Let $(u_i)$ be a sequence in $C^3(U)$ satisfying Hypotheses \ref{hypA},
\ref{hypB} and \ref{hyp:C_index_bound}, and $\vei \rightharpoonup V$.
Then $\dimh \sing V \leq n-7$ and
\begin{enumerate}[font = \upshape, label = (\roman*)]
	\item $\lambda_p(W) \geq \limsup_{i \to \infty} \lambda_p^i(W)$
	for all open $W \subset \subset U \setminus \sing V$ with
	$W \cap \reg V \neq \emptyset$ and all $p \in \NN$,
	\label{item:main_thm_spec}
	\item $\ind B_V \leq k$. \label{item:main_thm_ind_reg}
\end{enumerate}
\end{customthm}
\begin{rem}
The spectral lower bound remains true
if the assumptions are weakened and one assumes that for some $k \in \NN$
there is $\mu \in \RR$ such that
\begin{equation}
\lambda_k^i(U) \geq \mu \quad \text{for all $i \in \NN$}
\end{equation}
instead of an index bound---this observation is inspired
the work of Ambrozio--Carlotto--Sharp~\cite{AmbrozioCarlottoSharp2015}, where
a similar generalisation is made in the context of minimal surfaces.
One obtains the spectral bound via an inductive argument on $k$ similar to the argument
in Section~\ref{sec:proof_main_thm}, noting for the base case of
the induction that
bounds as in Corollary~\ref{cor:L2_bound_for_Ae_and_Be_if_u_stable}
hold if $\lambda_1^i \geq \mu$.
\end{rem}
The following corollary is an immediate consequence of
Theorem~\ref{thm:main_thm_propagation_of_index_bounds}.
\begin{customcor}{B}
\label{cor:main_thm_morse}
If $\reg V$ is two-sided, then its Morse index with respect to the
area functional satisfies $\ind_{\calH^n} \reg V \leq k$.
\end{customcor}
If $V$ is the stationary varifold arising from Guaraco's $1$-parameter
min-max construction~\cite{Guaraco15} (resp.\ from the $k$-parameter min-max construction of Gaspar--Guaraco~\cite{GasparGuaraco2016})
and its regular part is two-sided, then by Corollary~\ref{cor:main_thm_morse}
its Morse index is at most $1$ (resp.\ at most $k$).

\section{Preliminary results}
\label{section:preliminary_results}
The preliminary results are divided into three parts. In the first,
following~\cite{Tonegawa05}
we introduce `second fundamental forms' $A^i$ for the varifolds $V^i$
and relate them to the second variation of the Allen--Cahn functional.
The last two sections are dedicated to the spectra of the operators
$L_V = \Delta_V + \abs{A}^2 + \Ric_M(\nu,\nu)$ and $L_i = \Delta - \eps_i^{-2} W''(u_i)$.

\subsection{Stability and \texorpdfstring{$L^2$}{L2}--bounds of curvature}
\label{subsec:generalised_second_ff}
To simplify the discussion fix for the moment a
critical point $u \in C^3(U) \cap L^\infty(U)$ of the Allen--Cahn functional
$E_\eps$, with associated varifold $V^\eps$
defined by \eqref{eq:definition_varifolds}.

Let $x \in U$ be a regular point of $u$, that is $\nabla u(x) \neq 0$.
In a small enough neighbourhood
of $x$, the level set $\{ u = u(x)\}$ is embedded in $M$.
Call $\Sigma \subset M$ this embedded portion of the
level set, and let $A^\Sigma$ be its second fundamental form.
We use this to define a `second fundamental form' for $V^\eps$.
\begin{defn}
\label{defn:second_ff}
The function $A^\eps$ is defined at all $x \in U$ where $\nabla u(x) \neq 0$
by $A^\eps(x) = A^\Sigma(x)$
\end{defn}
\begin{rem}
Second fundamental forms can be generalised to the context
of varifolds via the integral identity \eqref{eq:gen_second_ff}---see
Appendix~\ref{app:second_fundamental_form_coordinate_expression}, or
\cite{Hutchinson86} for the original account of this theory.
Strictly speaking it is an abuse of language to call
$A^\eps$ the `second fundamental form' of
$V^\eps$, as it satisfies this identity only up to a small error term
\eqref{eq:approximate_identity_second_fundamental_form}.
\end{rem}

By definition $\nabla_X Y = \nabla^\Sigma_X Y + A^\Sigma(X,Y)$ for all
$X,Y \in C^1(T\Sigma)$.
Making implicit use of the musical isomorphisms here and throughout the text,
write $\nu^\eps(x) = \frac{\nabla u(x)}{\abs{\nabla u(x)}}$, so that
\begin{equation}
A^\Sigma(X,Y) = \langle \nabla_X Y,\nu^\eps \rangle \nu^\eps
= - \langle Y, \nabla_X \nu^\eps \rangle \nu^\eps.
\end{equation}
\begin{lem}
	\label{lem:second_ff_ae_bound}
	Let $x \in U$ be a regular point of $u$.
	Then
	\begin{equation}
	\label{eq:ae_bounded_hessian}
	\abs{\Ae}(x)^2
	\leq \frac{1}{\abs{\nabla u}^2(x)}
	(\abs{\nabla^2 u}^2(x) - \abs{\nabla \abs{\nabla u}}^2(x)),
	\end{equation}
	where $\nabla^2 u(x)$ is the Hessian of $u$ at $x$.
\end{lem}
\begin{proof}
	The second fundamental form $A^\Sigma$ is expressed in terms of the
	covariant derivative $\nabla \nu^\eps$ by
	\begin{equation}
	A^\Sigma = - \srestr{\nabla \nu^\eps}{T\Sigma \otimes T\Sigma} \otimes \nu^\eps.
	\end{equation}
	We can express $\nabla \nu^\eps$ as
	\begin{equation}
	\nabla \nu^\eps =
	\frac{\nabla^2 u}{\abs{\nabla u}} -
	\nu^\eps \otimes \frac{\nabla \abs{\nabla u}}{\abs{\nabla u}},
	\end{equation}
	whence after restriction to $T\Sigma \otimes T\Sigma$ we get
	\begin{equation}
	A^\Sigma =
	- \frac{1}{\abs{\nabla u}}
	\srestr{\nabla^2 u}{T\Sigma \otimes T\Sigma}
	\otimes \nu^\eps.
	\end{equation}
	On the other hand $\nabla \abs{\nabla u} = \langle \nabla^2 u, \nu^\eps \rangle$
	where $\nabla u \neq 0$, so after decomposing the Hessian $\nabla^2 u$
	in terms of its action on $T\Sigma$ and $N\Sigma$, we obtain
	\begin{equation}
	\abs{\nabla^2 u}^2 - \abs{\nabla \abs{\nabla u}}^2
	= \abs{\nabla u}^2 \abs{A^\Sigma}^2 +
	\abs{\srestr{\nabla^2 u}{T\Sigma \otimes N\Sigma}}^2
	\geq \abs{\nabla u}^2 \abs{A^\eps}^2. \qedhere
	\end{equation}
\end{proof}

When considering the
second variation, it somewhat simplifies notation to rescale the energy as
$\calE_\eps = \eps^{-1} E_\eps$.
Its second variation is
$ \delta^2 \calE_\eps(u)(\phi,\phi) =
\int_U \abs{\nabla \phi}^2 + \frac{W''(u)}{\eps^2} \phi^2, $
defined for all $\phi \in C_c^1(U)$, which by a density argument can be extended to $H_0^1(U)$.
The following identity will be useful throughout; a proof can be found
in either of the indicated sources.
\begin{lem}[\cite{Farina13,Tonegawa05}]
\label{lem:expr_second_variation}
Let $u \in C^3(U) \cap L^\infty(U)$ be a critical point of $E_\eps$.
For all $\phi \in C_c^1(U)$,
\begin{multline}
	\label{eq:second_variation_rescaled_expression}
	\delta^2 \calE_\eps(u)(\abs{\nabla u} \phi,\abs{\nabla u} \phi)
	= \\	\int_{U} \abs{\nabla u}^2 \abs{\nabla \phi}^2
	- \left(\abs{\nabla^2 u}^2 -
	\abs{\nabla \abs{\nabla u}}^2 + \Ric_M(\nabla u,\nabla u) \right)
	\phi^2 \dH^{n+1}.
\end{multline}
\end{lem}
Combining \eqref{eq:second_variation_rescaled_expression}
with the $\nve$-a.e. bound \eqref{eq:ae_bounded_hessian} yields
for all $\phi \in C_c^1(U)$
\begin{equation}
\label{eq:second_inequality_stability}
\frac{\eps}{2 \sigma}
\delta^2 \calE_{\eps}(u)(\abs{\nabla u}\phi,\abs{\nabla u}\phi)
\leq
\int \abs{\nabla \phi}^2
- (\abs{A^\eps}^2 + \Ric_M(\nu^\eps,\nu^\eps)) \phi^2 \dnve.
\end{equation}
When $u$ is stable, that is when $\delta^2 \calE_\eps(u)$ is non-negative,
then this identity yields $L^2(V^\eps)$--bounds for $A^\eps$.
\begin{cor}
	\label{cor:L2_bound_for_Ae_and_Be_if_u_stable}

	There is a constant $C = C(M) > 0$ such that if $u \in C^3(U) \cap L^\infty(U)$
	is a critical point of $E_\eps$ and is stable in an open ball $B(x,r) \subset U$
	of radius $r \leq 1$ then
	\begin{equation}
	\label{eq:bound_in_cor_stability_in_balls}
	\int_{B(x,\frac{r}{2})} \abs{\Ae}^2  \dnve
	\leq \frac{C}{r^2}  \norm{\ve}(B(x,r)).
	\end{equation}
\end{cor}
\begin{proof}
	The Ricci curvature
	term in \eqref{eq:second_inequality_stability} can be bounded by some
	constant $C(M) \geq 1$ as the ambient manifold $M$ is closed, so
	$ \int_{B(x,r)} \abs{\Ae}^2 \phi^2  \dnve
	\leq C(M) \int \phi^2 + \abs{\nabla \phi}^2 \dnve $
	for all $\phi \in C_c^1(B(x,r))$. Plug in a cut-off function
	$\eta \in C_c^1(B(x,r))$ with $\eta = 1$ in $B(x,\frac{r}{2})$ and
	$\abs{\nabla \eta} \leq 3r^{-1}$ to obtain the desired inequality.
\end{proof}
We now turn to a sequence $(u_i)$ of critical points satisfying
Hypotheses~\ref{hypA}--\ref{hyp:C_index_bound}.
If the $u_i$ are stable in a ball as in
Corollary~\ref{cor:L2_bound_for_Ae_and_Be_if_u_stable},
then the uniform weight bounds \eqref{eq:density_weight_measure}
imply uniform $L^2(V^i)$--bounds of the second fundamental forms,
which we denote $A^i$ from now on.
Under these conditions the $A^i$ converge
weakly to the second fundamental form $A$ (in the classical, smooth
sense) of $\reg V$.
\begin{prop}
	\label{prop:weak_convergence_second_ff}
	Let $W \subset \subset U \setminus \sing V$ be open with
	$W \cap \reg V \neq \emptyset$.
	If $\sup_i \int_W \abs{\Aei}^2 \dnvi < +\infty$,
	then passing to a subsequence $\Aei \dvei \rightharpoonup A \dv$ weakly as Radon
	measures on $G_n(W)$, and
	\begin{equation}
	\label{eq:bound_fatou_second_ff}
	\int_W \abs{A}^2 \dnv \leq
	\liminf_{i \to \infty} \int_W \abs{\Aei}^2 \dnvi,
	\end{equation}
	where $A$ is the second fundamental form of $\reg V \subset M$.
\end{prop}

The weak subsequential convergence follows immediately from compactness
of Radon measures; the main difficulty is to show that the weak limit is $A \dv$.
The proof is a straight-forward adaptation of the argument
used for the stable case in~\cite{Tonegawa05}; we present a
complete argument in Appendix~\ref{app:second_fundamental_form_coordinate_expression}
for the reader's convenience.

\subsection{Spectrum of \texorpdfstring{$L_V$}{LV} and weighted min-max}
\label{sec:prelim_spectrum_sec_var}

Throughout we restrict ourselves to a compactly contained open
subset $W \subset \subset U \setminus \sing V$
to avoid blow-up of the coefficients of $L_V$ near the singular set,
and assume $W \cap \reg V \neq \emptyset$ to avoid vacuous statements.
As $W \cap \reg V$ is compactly contained in $\reg V$, it can
intersect only finitely many connected components $C_1,\dots,C_N$
of $\reg V$.
By the constancy theorem~\cite[Thm.~41.1]{Simon84}
the multiplicity function $\Theta$ of a stationary
integral varifold $V$ is constant on every connected component of $\reg V$;
we write $\Theta_1,\dots,\Theta_N$ for the respective multiplicities
of $C_1,\dots,C_N$.

By classical theory for elliptic PDE~\cite[Ch.~8]{GilbargTrudinger98},
the spectrum of $L_V$ has the following min-max characterisation:
\begin{equation}
\label{eq:unw_min_max}
\lambda_p = \inf_{\dim S = p} \max_{\phi \in S \setminus \{ 0 \}}
\frac{B_{\reg V}(\phi,\phi)}{\norm{\phi}^2_{L^2}}
\quad \text{for all $p \in \NN$},
\end{equation}
where the infimum is taken over linear subspaces $S$
of $H_0^1$ (recall this is our
abbreviated notation for $H_0^1(W \cap \reg V)$).
From this we easily obtain a min-max characterisation that is `weighted' by
the multiplicities  $\Theta_1,\dots,\Theta_N$ in the sense that
\begin{equation}
\label{eq:w_min_max}
\lambda_p = \inf_{\dim S = p} \max_{\phi \in S \setminus \{0 \}}
\frac{B_V(\phi,\phi)}{\norm{\phi}^2_{L^2(V)}}
\quad \text{for all $p \in \NN$}.
\end{equation}
To see this, observe the following: as functions $\phi \in H_0^1$
vanish near the boundary of every connected component $C \subset \reg V$,
the function $\phi_C$ on $W \cap \reg V$ defined by
\begin{equation}
\phi_C = \begin{cases}
\phi &\text{on }  C \\
0 & \text{on } W \cap \reg V \setminus C
\end{cases}
\end{equation} also belongs to $H_0^1$.
Moreover
\begin{equation}
B_V(\phi_C,\phi_C)
= \Theta_C B_{\reg V}(\phi_C,\phi_C)
\text{ and }
\norm{\phi_C}^2_{L^2(V)}
= \Theta_C \norm{\phi_C}^2_{L^2},
\end{equation}
where $\Theta_C$ denotes the multiplicity of $C$.
We then define a linear isomorphism of $H_0^1$ via normalisation
by the respective multiplicities of the components. This sends
$\phi \mapsto \bar{\phi} := \sum_{j=1}^N \Theta_j^{-1/2} \phi_{C_j}$; then
\begin{equation}
\frac{B_V(\bar{\phi},\bar{\phi})}{\norm{\bar{\phi}}_{L^2}^2}
= \frac{B_{\reg V}(\phi,\phi)}{\norm{\phi}_{L^2(V)}^2}.
\end{equation}
Therefore the `unweighted' and `weighted' min-max
characterisations \eqref{eq:unw_min_max} and \eqref{eq:w_min_max} are in fact equivalent.
In the remainder we mainly use \eqref{eq:w_min_max}, and
abbreviate this as $\lambda_p = \inf_{\dim S = p} \max_{S \setminus \{0\}} J_V$,
where $J_V$ denotes the `weighted' Rayleigh quotient
\begin{equation}
J_V(\phi) = \frac{B_V(\phi,\phi)}{\norm{\phi}_{L^2(V)}^2}
\text{\quad for all $\phi \in H_0^1 \setminus \{0\}$}.
\end{equation}

The min-max characterisation implies the following lemma, which
highlights the dependence of the spectrum $\lambda_p(W)$ on
the subset $W$.
\begin{lem}
	\label{lem:shrinking_balls_spectrum}
	\leavevmode

	\begin{enumerate}[label = (\alph{*}), font = \upshape]
		\item If $W_1 \subset W_2 \subset \subset U \setminus \sing V$, then
		$\lambda_p(W_1) \geq \lambda_p(W_2)$: the spectrum
		is monotone decreasing.
		\item \label{item:index_superadd} If $W_1,W_2 \subset \subset U \setminus \sing V$
		have $W_1 \cap W_2 = \emptyset$, then
		$\ind_{W_1} B_V + \ind_{W_2} B_V = \ind_{W_1 \cup W_2} B_V$.

		\item \label{item:shrinking_balls}
		If $W \subset \subset U \setminus \sing V$ and $y \in W \cap \reg V$, then
		$\lambda_p(W) = \lim_{R \to 0} \lambda_p(W \setminus \overline{B}(y,R))$.
	\end{enumerate}

\end{lem}
\begin{rem}
	The same properties hold for the spectrum and index of $L_i$, and the proof is easily
	modified to cover this case.
\end{rem}
\begin{proof}
(a) This is immediate from the min-max characterisations,
or simply by definition of the spectrum. Similarly for \ref{item:index_superadd}.

\ref{item:shrinking_balls} By monotonicity of the spectrum we have
\begin{equation}
\lambda_p(W \setminus \overline{B}(y,R))
\geq \lambda_p(W \setminus \overline{B}(y,R'))
\geq \lambda_p(W)
\end{equation}
for all $R > R' > 0$. The limit as $R \to 0$ therefore exists and is bounded below
by $\lambda_p(W)$; it remains only to show that $\lim_{R \to 0}
\lambda_p(W \setminus \overline{B}(y,R)) \leq \lambda_p(W)$.

By monotonicity of the spectrum it is equivalent to show that
for a fixed radius $R > 0$, $\lim_{m \to \infty}
\lambda_p(W \setminus \overline{B}(y,2^{-m}R))
\leq \lambda_p(W)$.
Let $(\rho_m)_{m \in \NN}$ be a sequence
in $C_c^1(B(y,R) \cap \reg V)$ with the following properties (such a sequence
exists provided $n \geq 2$, see Remark~\ref{rem:cutoff_functions} below):
\begin{enumerate}[label = (\arabic*)]
	\item \label{item:cutoff_props_a} $\srestr{\rho_m}{B(y,2^{-m}R) \cap \reg V}
	\equiv 0$ and 	$\rho_m \to 1$ $\calH^n$-a.e.\ in
	$W \setminus \{ y \} \cap \reg V $,

	\item \label{item:cutoff_props_b}
	$\norm{\nabla_V \rho_m}_{L^2(W \cap \reg V)} \to 0$.
\end{enumerate}
Let a small $\delta > 0$ be given and choose a family $(\phi_1,\dots,\phi_p)$
in $C_c^1(W \cap \reg V)$ whose $\linspan(\phi_1,\dots,\phi_p) =: S$ has
$\max_{S \setminus 0} J_V \leq \lambda_p(W) + \delta$.
Write $\rho_m S$ for
$\linspan(\rho_m \phi_1,\dots,\rho_m \phi_p)\subset
C_c^1(W \setminus \overline{B}(y,2^{-m}R) \cap \reg V)$---for
$m$ large enough the functions $\rho_m \phi_i$ are indeed linearly independent.
By the weighted min-max formula \eqref{eq:w_min_max},
\begin{equation}
\max_{\rho_m S \setminus 0} J_V
\geq \lambda_p(W \setminus \overline{B}(y,2^{-m}R)).
\end{equation}

Let $t_m \in \Sph^{p-1} \subset \RR^p$ denote the coefficients of the linear combination
$t_m \cdot \rho_m \phi := \rho_m \sum_{j=1}^p t_{mj} \phi_j \in \rho_m S$
that realises $\max_{\rho_m S \setminus 0} J_V$.
Passing to a convergent subsequence $t_m \to t \in \Sph^{p-1} \subset \RR^p$
we get $J_V(t_m \cdot \rho_m \phi) \to J_V(t \cdot \phi)$, and hence
\begin{equation}
\lim_{m \to \infty} J_V(t_m \cdot \rho_m \phi)
\leq \max_{S \setminus 0} J_V.
\end{equation}
On the one hand $\max_{S \setminus 0} J_V \leq \lambda_p(W) + \delta$ by
our choice of $S$,
on the other hand $\lim_{m \to \infty} \lambda_p(W \setminus \overline{B}(y,2^{-m}R))
\leq \lim_{m \to \infty} J_V(t_m \cdot \rho_m \phi)$ by our choice of $t_m$.
The conclusion follows after combining these two observations and letting $\delta \to 0$.
\end{proof}

\begin{rem}
\label{rem:cutoff_functions}
A sequence of functions $(\rho_m)$ with properties
\ref{item:cutoff_props_a} and \ref{item:cutoff_props_b} exists provided $n \geq 2$,
as we assume throughout. When $n \geq 3$ one can use the standard cutoff functions;
for $n = 2$ a more precise construction is necessary, described
for instance in~\cite[Sec.~4.7]{Evans15}.
\end{rem}

\subsection{Spectrum of \texorpdfstring{$L_i$}{Li} and conditional proof of
	Theorem~\texorpdfstring{\ref{thm:main_thm_propagation_of_index_bounds}}{A}}
The main result in this section is Lemma~\ref{lem:theorem_holds_if_second_ff_bounded};
essentially it says that
\begin{equation}
\lambda_p(W) \geq \limsup_{i \to \infty} \lambda_p^i(W)
\end{equation}
holds under the condition that $\sup_i \int_W \abs{A^i}^2 \dnvi < +\infty$.
What precedes it in this section are technical results required for its proof.

Again, by classical elliptic PDE theory the eigenvalues $\lambda_p^i(W)$
of $L_i = \Delta - \eps_i^{-2} W''(u_i)$ on $H_0^1(W)$
have the following min-max characterisation
in terms of the rescaled Allen--Cahn functional
$\calE_{\eps_i} = \eps_i^{-1} E_{\eps_i}$:
\begin{equation}
\label{eq:min_max_L_i}
\lambda_p^i(W) = \inf_{\dim S = p} \max_{\phi \in S \setminus \{0\}}
\frac{\delta^2 \calE_{\eps_i}(u_i)(\phi,\phi)}{\norm{\phi}_{L^2}^2}
\quad \text{for all $p \in \NN$},
\end{equation}
where the infimum is over $p$-dimensional linear subspaces
$S \subset H_0^1(W)$.
Define the \emph{Rayleigh quotient} $J_i$ by
\begin{equation}
J_i(\phi) = \frac{\delta^2 \calE_{\eps_i}(u_i)(\phi,\phi)}{\norm{\phi}_{L^2}^2}
\quad \text{for all $\phi \in H_0^1(W) \setminus \{0\}$},
\end{equation}
so that we can write the min-max characterisation more succinctly as
$\lambda_p^i = \inf_{\dim S = p} \max_{S \setminus \{0\}} J_i.$

To compare the spectrum of $L_V$ in $H_0^1$ with those of the
operators $L_i$ in $H_0^1(W)$,
extend functions in $C_c^1(W \cap \reg V)$ to $C_c^1(W)$
in the standard way, which we now describe to fix notations.
Pick a small enough $0 < \tau < \inj(M)$ so that $B_\tau V :=
\exp N_\tau V$ is a tubular neighbourhood of $W \cap \reg V$, where
$N_\tau V := \{ s_p \in NV \mid p \in W \cap \reg V, \abs{s_p} < \tau \}$.
We abuse notation slightly to denote points in $B_\tau V$
by $s_p$, and identify the fibre $N_pV$ with
$(\exp_p)_* N_p V \subset T_{s_p} (B_\tau V)$.
The distance function $d_V: x \in B_\tau V \mapsto \dist(x,\reg V)$
is Lipschitz and smooth on $B_\tau V \setminus \reg V$.
By the Gauss lemma $\grad d_V(s_p) = -s_p/\abs{s_p}$ for all
$s_p \in B_\tau V \setminus \reg V$.
A function $\phi \in C^1(B_\tau V)$ is constant
along geodesics normal to $\reg V$ if $\phi(s_p) =
\phi(0_p)$ for all $s_p \in B_\tau V$,
or equivalently if $\langle \nabla \phi,\nabla d_V  \rangle \equiv 0$
in $B_\tau V \setminus \reg V$.

\begin{lem}
	\label{lem:extension_scalar_functions}
	Any $\phi \in C_c^1(W \cap \reg V)$ can be extended to $C_c^1(W)$
	with $\langle \nabla \phi, \nabla d_V \rangle \equiv 0$
	in $B_{\frac{\tau}{2}} V \setminus \reg V$ for some $\tau = \tau(\phi) > 0$.
\end{lem}
\begin{proof}
	Extend $\phi \in C_c^1(W \cap \reg V)$ to
	$B_\tau V$ by setting $\tilde{\phi}(s_p) = \phi(p)$,  so that
	$\langle \nabla d_V , \nabla \tilde{\phi} \rangle \equiv 0$ in
	$B_\tau V \setminus \reg V$.

	Let $\eta \in C^1[0,\infty)$ be a cutoff
	function with $0 \leq \eta \leq 1$, $\eta \equiv 1 $ on $[0,1/2)$
	and $\spt \eta \subset [0,1)$.

	Then
	\begin{equation}
	(\eta \circ d_V/\tau) \tilde{\phi} \in C_c^1(B_\tau V)
	\text{ and }  (\eta \circ d_V/\tau) \tilde{\phi} = \tilde{\phi} \text{ on } B_{\tau/2} V.

	\end{equation}
	Moreover even though $B_\tau V \not\subset W$ in general, as $\spt \phi$
	is compactly contained in $W \cap \reg V$ we still have
	$ (\eta \circ d_V/\tau) \tilde{\phi} \in C_c^1(W)$ provided
	$0 < \tau < \dist (\spt \phi, \partial W)$.
\end{proof}
The following lemma gives an asymptotic lower bound for the Rayleigh
quotient $J_V$ in terms of the $J_i$.
\begin{lem}
	\label{lem:min_max_convergence_when_C_c_convergence}
	Let $B_{\tau}V$
	be a tubular neighbourhood of $W \cap \reg V$ with width $\tau > 0$,
	and let $(\phi_i)_{i \in \NN}$ be a sequence of functions in $C_c^1(W)$ with

	\begin{enumerate}[label = (\alph*), font = \upshape]
		\item $\langle \nabla \phi_i,\nabla d \rangle \equiv 0$ in $W \cap B_{\tau/2}V$
		for all $i$,
		\item $\phi_i \to \phi$ in $C_c^1(W)$ as $i \to \infty$,
		where $\phi \neq 0$ in $W \cap \reg V$.
		\label{item:Cc_cvg_second_assumption}
	\end{enumerate}

	If $\sup_i \int_W \abs{A^i}^2 \dnvi < +\infty$,
	then $J_V(\phi) \geq \limsup_{i \to \infty}
	J_i(\abs{\nabla u_i}\phi_i)$.
\end{lem}
\begin{proof}
Before we start the proof proper, note that for all $\phi \in H_0^1(U)$,
dividing both sides of \eqref{eq:second_inequality_stability}
by $ \frac{\eps_i}{2 \sigma} \int \phi^2 \abs{\nabla u_i}^2 \dH^{n+1}
= \int \phi^2 \dnvi$ yields
\begin{equation}
\label{eq:sec_var_sec_form}
\norm{\phi}_{L^2(\vi)}^{-2}
\int \abs{\nabla \phi}^2 - (\abs{A^i}^2 + \Ric_M(\nui,\nui)) \phi^2 \dnvi
\geq J_i(\abs{\nabla u_i}\phi),
\end{equation}
provided of course that $\norm{\phi}_{L^2(V^i)}^2 \neq 0$.

We treat the terms on the left-hand side separately in the calculations
\eqref{item:proof_thmA_i_1}--\eqref{item:proof_thmA_i_4} below.
Once these are completed, we combine \eqref{item:proof_thmA_i_1} with our
assumption that $\norm{\phi}_{L^2(V)}^2 \neq 0$ to obtain that
$\norm{\phi_i}_{L^2(V^i)} \neq 0$ for large enough $i$.
The conclusion follows by combining \eqref{eq:sec_var_sec_form}
with the remaining calculations:
\begin{enumerate}
	\item $\int \phi^2 \dnv = \lim_{i \to \infty} \int \phi_i^2
	\dnvi$\label{item:proof_thmA_i_1}
	\item $\int \abs{\nabla_V \phi}^2 \dnv = \lim_{i \to \infty} \int
	\abs{\nabla \phi_i}^2 \dnvi$, \label{item:proof_thmA_i_2}

	\item $\int \abs{A}^2 \phi^2 \dnv
	\leq \liminf_{i \to \infty} \int \abs{A^i}^2 \phi_i^2 \dnvi$,
	\label{item:proof_thmA_i_3}
	\item $\int \Ric_M(\nu,\nu) \phi^2 \dnv
	= \lim_{i \to \infty } \int \Ric_M(\nui,\nui)\phi_i^2 \dnvi$.
	\label{item:proof_thmA_i_4}
\end{enumerate}

\eqref{item:proof_thmA_i_1}
By assumption $\phi_i^2 \to \phi^2$
in $C_c(W)$, whence by Corollary~\ref{cor:meas_fct_cvg} we get
$\int \phi^2 \dnv = \lim_{i \to \infty} \int \phi_i^2 \dnvi$.
The same argument proves \eqref{item:proof_thmA_i_2}, after noticing
that $\langle \nabla \phi_i, \nabla d_V \rangle \equiv 0$ implies
$\abs{\nabla \phi}^2 = \abs{\nabla_V \phi}^2$ on $W \cap \reg V$.

\eqref{item:proof_thmA_i_3}
The sequence $(A^i \phi_i \dnvi)$ converges weakly to $A \phi \dnv$,
as we can show by
testing against an arbitrary $\varphi \in C_c(U)$:
\begin{multline}
\int A^i \phi_i \varphi \dnvi
- \int A \phi \varphi \dnv
= \\ \int A^i (\phi_i - \phi) \varphi \dnvi
+ \int A^i \phi \varphi \dnvi - \int A \phi \varphi \dnv.
\end{multline}
The first integral is bounded by
\begin{equation}
\Big \lvert \int A^i (\phi_i - \phi) \varphi \dnvi \Big \rvert
\leq
\norm{\phi_i - \phi}_{L^\infty} \norm{\varphi}_{L^2(\vi)}
\norm{A^i}_{L^2(V^i)} \to 0
\end{equation}
because $\phi_i \to \phi$
in $C_c(W)$ as $i \to \infty$.
The remaining terms tend to $0$ by the weak convergence of
$A^i \dnvi \rightharpoonup A \dnv$ tested against
$\phi \varphi \in C_c(W)$.
Then inequality \eqref{eq:measure_function_convergence_L2_bound}
gives
$
\int \abs{A}^2 \phi^2 \dnv
\leq \liminf_{i \to \infty} \int \abs{A^i}^2 \phi_i^2 \dnvi.
$

\eqref{item:proof_thmA_i_4}
For each $S \in G_n(T_p M)$ pick a unit vector $\nu_S$ in $T_p M$
orthogonal to $S$, and define a smooth function $R_M$ on $G_n(U)$ by
$R_M \! : S \mapsto \Ric_M(\nu_S,\nu_S)$.
Then $\phi_i^2 R_M \to \phi^2 R_M$ in $C_c(G_n(U))$ as $i \to \infty$,
and by Corollary~\ref{cor:meas_fct_cvg},
\begin{multline}
\int  \phi_i^2 \Ric_M(\nui,\nui) \dnvi
= \\  \int \phi_i^2 R_M \dvi
\to
\int \phi^2 R_M \dv = \int \phi^2 \Ric_M(\nu,\nu) \dnv.
\qedhere
\end{multline}
\end{proof}

We conclude the section with a proof of
Theorem~\ref{thm:main_thm_propagation_of_index_bounds}\ref{item:main_thm_spec}
in the case where there is a uniform $L^2(\vei)$--bound on the second
fundamental forms $(\Aei)$.
\begin{lem}
	\label{lem:theorem_holds_if_second_ff_bounded}
	Let $W \subset \subset U \setminus \sing V$ be open with
	$W \cap \reg V \neq \emptyset$.
	If $\sup_i \int_W \abs{\Aei}^2 \dnvi < +\infty$,
	then $\lambda_p(W) \geq \limsup_{i \to \infty} \lambda_p^i(W)$
	for all $p$.
\end{lem}
\begin{proof}
We may assume that every connected component of $W$ intersects $\spt \nv$
(or $\reg V$, equivalently as $W \cap \sing V = \emptyset$)
without restricting generality: if $C$ is a
connected component of $W$ with $C \cap \reg V = \emptyset$,
then $\lambda_p(W \setminus C) = \lambda_p(W)$, and by monotonicity
$\lambda_p^i(W \setminus C) \geq \lambda_p^i(W)$ for all $i$.

Given $\delta > 0$ there is a $p$-dimensional linear subspace
$S = \linspan(\phi_1,\dots,\phi_p)$ of $C_c^1(W \cap \reg V)$ with
\begin{equation}
\label{eq:almost_maximiser_but_in_C_c}
\lambda_p(W) + \delta
\geq
\max_{S \setminus \{0\}} J_V.
\end{equation}
Extend the functions $\phi_i$
to $C_c^1(W)$ as in Lemma~\ref{lem:extension_scalar_functions};
for large enough $i$ the family
$(\abs{\nabla u_i}\phi_1,\dots,\abs{\nabla u_i} \phi_p)$
is still linearly independent.
Indeed, otherwise we could extract a subsequence
such that $(\abs{\nabla u_{i'}} \phi_1,\dots,
\abs{\nabla u_{i'}} \phi_p)$ has a linear dependence, with coefficients
$a_{i'} \neq 0 \in \RR^p$ say.
Then notice that
\begin{equation}
\abs{\nabla u_{i'}} a_{i'} \cdot  \phi = 0
\Leftrightarrow \norm{a_{i'} \cdot \phi}_{L^2(V^{i'})} = 0,
\end{equation}
where we abbreviated
$a_{i'} \cdot \phi := \sum_{j=1}^p a_{i'j} \abs{\nabla u_{i'}} \phi_j$.
We may normalise the coefficients $a_{i'}$ so as to guarantee
$\abs{a_{i'}} = 1$ and then, possibly after extracting a second subsequence, assume that $a_{i'} \to a \in \Sph^{p-1}$ as $i' \to \infty$.
The resulting strong convergence
$a_{i'} \cdot \phi \to a \cdot \phi$ in $C_c(W)$
combined with $\nvi \rightharpoonup \nv$ yield
$\norm{a_{i'} \cdot \phi}_{L^2(V^{i'})} \to \norm{a \cdot \phi}_{L^2(V)}$;
this contradicts $\norm{a \cdot \phi}_{L^2(V)} > 0$.

From now on take $i$ large enough so that
$(\abs{\nabla u_i} \phi_1,\cdots,\abs{\nabla u_i} \phi_p)$
is linearly independent.
For such large $i$, we may let $t_i \in \Sph^{p-1} \subset \RR^p$
be the (normalised) coefficients of a linear combination
$t_i \cdot \abs{\nabla u_i} \phi
= \sum_{j=1}^p t_{ij} \abs{\nabla u_i} \phi_j$
that maximises the Rayleigh quotient $J_i$:
\begin{equation}
J_i(t_i \cdot \abs{\nabla u_i} \phi)
= \max_{\abs{\nabla u_i} S \setminus \{0\}} J_i \geq \lambda_p^i(W).
\end{equation}
Extract a convergent subsequence $t_{i'} \to t \in \Sph^{p-1}$,
so that $t_{i'} \cdot \phi \to t \cdot \phi$ in $C_c^1(W)$ as $i' \to \infty$.
Lemma~\ref{lem:min_max_convergence_when_C_c_convergence} gives
$J_V(t \cdot \phi) \geq \limsup_{i \to \infty}
\max_{\abs{\nabla u_i} S\setminus \{ 0\}} J_i,$
which in turn is greater than $\limsup_{i \to \infty} \lambda_p^{i}(W)$.
Using \eqref{eq:almost_maximiser_but_in_C_c},
\begin{equation}
\lambda_p(W) + \delta
\geq J_V(t \cdot \phi)
\geq \limsup_{i \to \infty} \lambda_p^i(W),
\end{equation}
and we conclude by letting $\delta \to 0$.
\end{proof}

Lemma~\ref{lem:theorem_holds_if_second_ff_bounded}
has the following immediate corollary.
\begin{cor}
	\label{cor:thmA_ii_when_second_ff_is_bounded}
	Under the hypotheses of Lemma~\ref{lem:theorem_holds_if_second_ff_bounded},
	\begin{equation}
	\ind_W B_V \leq \liminf_{i \to \infty} (\ind_W u_i).
	\end{equation}
\end{cor}

\section{Proof of the main theorem (Theorem A)}
\label{sec:proof_main_thm}
We briefly recall the context of the proof: $M^{n+1}$ is a closed Riemannian
manifold and $U \subset M$ is an arbitrary open subset. The sequence
of functions $(u_i)$ in $C^3(U)$ satisfies Hypotheses \ref{hypA}, \ref{hypB}
and \ref{hyp:C_index_bound}---the last hypothesis says that
$\ind u_i \leq k$ for all $i$.
To every $u_i$ we associate the $n$--varifold $\vei$ from
\eqref{eq:definition_varifolds}. By Theorem~\ref{lem:properties_V},
we may pass to a subsequence of $(\vei)$ converging
weakly to a stationary integral varifold $V$.
\subsection{Spectrum and index of \texorpdfstring{$V$}{V}: proof of (i) and (ii)}
The main idea, inspired by an argument of Bellettini--Wickramasekera
\cite{BelWic16}, is to fix a compactly contained open subset
 $W \subset \subset U \setminus \sing V$ and study
the stability of $u_i$ in open balls covering $W \cap \reg V \neq \emptyset$.
We then shrink the radii of the covering balls to $0$, and prove
the spectral lower bound of
Theorem~\ref{thm:main_thm_propagation_of_index_bounds}\ref{item:main_thm_spec} by
induction on $k$. The upper bound on $\ind B_V$ of
Theorem~\ref{thm:main_thm_propagation_of_index_bounds}\ref{item:main_thm_ind_reg}
is then an immediate consequence.

In the base of induction the $u_i$ are stable in $U$.
Let $\eta \in C_c^1(U)$ be a cutoff function constant equal to $1$ on $W$.
The stability inequality \eqref{eq:second_inequality_stability} gives that
\begin{equation}
\int_W \abs{\Aei}^2 \dnvi
\leq C(M) \dist(W,\partial U)^{-2}
\norm{\vei}(U) \quad \text{for all $i$}.
\end{equation}
Combining this with \eqref{eq:density_weight_measure}
we get $\sup_i \int_W \abs{\Aei}^2 \dnvi < +\infty$,
and thus $\lambda_p(W) \geq \limsup_{i \to \infty} \lambda_p^i (W)$
by Lemma~\ref{lem:theorem_holds_if_second_ff_bounded}.

For the induction step, let $k \geq 1$ and assume that
Theorem~\ref{thm:main_thm_propagation_of_index_bounds}\ref{item:main_thm_spec}
holds with $k-1$ in place of $k$. Consider an arbitrary $W \subset \subset
U \setminus \sing V$ that intersects $\reg V$.
Fix a radius $0 < r < \dist(W,\sing V)$, and pick points
$x_1,\dots,x_N \in W \cap \reg V$ such that
$W \cap \reg V \subset \cup_{j = 1}^N B(x_j,\frac{r}{2})$.
We define the following \emph{Stability Condition} for the cover
$\{B(x_j,\frac{r}{2})\}_{1 \leq j \leq N}$:
\begin{enumerate}[label={(SC)}]
	\item \emph{For large $i$, each $u_i$ is stable in every ball
		$B(x_1,r),\dots,B(x_N,r)$. \label{item:stability_condition}}
\end{enumerate}

\begin{claim}
\label{claim:SH_regularity}
If for the cover $\{B(x_j,\frac{r}{2})\}_{1 \leq j \leq N}$:
\begin{enumerate}[label = (\alph*)]
\item \label{item:reg_SH_holds}
\ref{item:stability_condition} holds,
then $\lambda_p(W) \geq \limsup_{i \to \infty} \lambda_p^i(W)$;

\item \label{item:reg_SH_fails}
\ref{item:stability_condition} fails, then $\lambda_p(W \setminus \overline{B_r})
\geq \limsup_{i \to \infty} \lambda_p^i(W \setminus \overline{B_r})$
for some ball $B_r\in \{ B(x_j,r)\}$.
\end{enumerate}
\end{claim}

\begin{proof}
\ref{item:reg_SH_holds}
Let $W_r = W \cap \cup_{j=1}^N B(x_j,\frac{r}{2})$, so that
$W_r \cap \reg V = W \cap \reg V$ and hence $\lambda_p(W_r) = \lambda_p(W)$.
Moreover $W_r \subset W$, so $\lambda_p^i(W_r) \geq \lambda_p^i(W)$
for all $i$ by monotonicity of the spectrum.
Therefore it is enough to show that $\lambda_p(W_r)
\geq \limsup_{i \to \infty} \lambda_p^i(W_r)$.

Because \ref{item:stability_condition} holds, summing
\eqref{eq:bound_in_cor_stability_in_balls} over all balls we get
\begin{equation}\label{bound}
\int_{W_r} \abs{\Aei}^2 \dnvi \leq
\frac{NC}{r^2} \norm{\vei}(W_r) \leq \frac{NCE_{0}}{2 r^2 \sigma}
\quad \text{for all $i$},
\end{equation}
so $\lambda_p(W_r) \geq \limsup_{i \to \infty} \lambda_p^i(W_r)$
by Lemma~\ref{lem:theorem_holds_if_second_ff_bounded}.

\ref{item:reg_SH_fails} If \ref{item:stability_condition} fails, then
some subsequence $(u_{i'})$ must be unstable in one of the balls
$B_r \in \{B(x_j,r)\}$,
in other words $\ind_{B_r} u_{i'} \geq 1$ for all $i'$.
On the other hand
\begin{equation}
\ind_{B_r} u_{i'} + \ind_{W \setminus \overline{B_r}} u_{i'}
\leq \ind_W u_{i'}
\end{equation}
because $B_r$ and $W \setminus \overline{B_r}$ are disjoint
open sets.
As $\ind_W u_{i'} \leq k$, we get
$\ind_{W \setminus \overline{B_r}} u_{i'} \leq k-1$
for all $i'$, and we conclude after applying the
induction hypothesis to $(u_{i'})$ in $W \setminus \overline{B_r}$.\end{proof}
\begin{rem}
This argument shows that when \ref{item:stability_condition} fails there is
a ball $B_r\in \{B(x_j,r)\}$ with $\lambda_p(W \setminus \overline{B_r})
\geq \limsup_{i \to \infty} \lambda_{p+1}^i(W)$ for $p \geq k$, and thus also
$\ind_{W \setminus \overline{B_r}} B_V \leq k-1$, but
the induction step only requires the weaker conclusion
from Claim~\ref{claim:SH_regularity}.
\end{rem}

Now consider a decreasing sequence $r_m \to 0$ with
$0 < r_m < \dist(W,\sing V)$ and reason as above with $r = r_m$.
For each $m$, pick points $x_1^m,\dots,x_{N_{m}}^m \in W \cap \reg V$
such that
\begin{equation}
W \cap \reg V \subset \cup_{j=1}^{N_m} B \big(x_j^m,\frac{r_m}{2} \big).
\end{equation}
If \ref{item:stability_condition} holds for a cover
$\{B(x_j^m,\frac{r_m}{2})\}$, then
$\lambda_p(W) \geq \limsup_{i \to \infty} \lambda_p^i(W)$
by Claim~\ref{claim:SH_regularity}, and the induction step
is completed.

Otherwise \ref{item:stability_condition} fails for all constructed covers,
and by Claim~\ref{claim:SH_regularity} there is a sequence $(y_m)$ in
$W \cap \reg V$ with
\begin{equation}
\lambda_p (W \setminus \overline{B}(y_m,r_m))
\geq \limsup_{i \to \infty} \lambda_p^i(W \setminus \overline{B}(y_m,r_m)),
\end{equation}
and thus by monotonicity of the spectrum
\begin{equation}
\label{eq:spec_lower_bound_outside_ball}
\lambda_p (W \setminus \overline{B}(y_m,r_m)) \geq
\limsup_{i \to \infty} \lambda_p^i(W).
\end{equation}
Passing to a subsequence if necessary, we may assume that
$(y_m)$ converges to a point $y \in \overline{W} \cap \reg V$.
If we fix a radius $R > 0$, then $B(y_m,r_m) \subset B(y,R)$ for large enough $m$,
so by monotonicity and \eqref{eq:spec_lower_bound_outside_ball},
\begin{equation}
\lambda_p(W \setminus \overline{B}(y,R))
\geq \limsup_{m \to \infty} \lambda_p(W \setminus \overline{B}(y_m,r_m))
\geq \limsup_{i \to \infty} \lambda_p^i(W).
\end{equation}
The conclusion follows after combining this with
$\lambda_p(W) = \lim_{R \to 0} \lambda_p(W \setminus \overline{B}(y,R))$
from Lemma~\ref{lem:shrinking_balls_spectrum}.

Together with the base of induction, we have proved
Theorem~\ref{thm:main_thm_propagation_of_index_bounds}\ref{item:main_thm_spec}
for all sequences $(u_i)$ with $\sup_i \ind u_i \leq k$ for some $k \in \NN$.
The index bound $\ind_W B_V \leq k$ follows immediately:
the spectral lower bound implies that $L_V$ must have fewer
negative eigenvalues than the $L_i$ as $i \to \infty$. Therefore
\begin{equation}
\ind_W B_V = \mathrm{card} \{p \in \NN \mid \lambda_p(W)  < 0 \} \leq k.
\end{equation}
As the subset $W$ was arbitrary we get $\ind B_V \leq k$;
this proves
Theorem~\ref{thm:main_thm_propagation_of_index_bounds}\ref{item:main_thm_ind_reg}.

\subsection{Regularity of \texorpdfstring{$V$}{V}: proof of \texorpdfstring{$\dimh \sing V \leq n-7$}{dimh sing V <= n-7}}
\label{subsection:proof_regularity_of_v}
The approach is the same as in the proof of
Theorem~\ref{thm:main_thm_propagation_of_index_bounds}\ref{item:main_thm_spec}--\ref{item:main_thm_ind_reg}
with one difference: we again proceed by induction on $k$, but
we now cover the entire support $\spt \nv$ (including the singular set),
instead of constructing covers a positive distance away from $\sing V$.

The base of induction, where the $u_i$ are stable in $U$,
was proved in~\cite{TonegawaWickramasekera10}.

For the induction step, suppose that $\dimh (U' \cap \sing V) \leq n-7$
holds with $k-1$ in place of $k$, and for arbitrary open subsets $U' \subset U$.
Fix $r > 0$, and choose points $x_1,\dots,x_N \in U \cap \spt \norm{V}$
such that $U \cap \spt \norm{V} \subset \cup_{j = 1}^N B(x_j,r)$.
The \emph{Stability Condition} for the cover
$\{B(x_j,r) \}_{1 \leq j \leq N}$ is defined in the same way as above,
except the radii need not be doubled:
\begin{enumerate}[label={(SC)}]
\item \emph{For large $i$, each $u_i$ is stable in every ball
	$B(x_1,r),\dots,B(x_N,r)$. \label{item:stability_condition_reg}}
\end{enumerate}
\begin{claim}
\label{claim:reg_SC_holds}
If for the cover $\{ B(x_j,r) \}_{1 \leq j \leq N}$:
\begin{enumerate}[label = (\alph*)]
	\item \label{item:reg_SC_holds} \ref{item:stability_condition_reg} holds,
	then $\dimh \sing V \leq n-7$;
	\item \label{item:reg_SC_fails} \ref{item:stability_condition_reg} fails,
	then $\dimh \sing V \setminus \overline{B_r} \leq n-7$
	for some ball $B_r\in \{B(x_j,r)\}$.
\end{enumerate}
\end{claim}
\begin{proof}
\ref{item:reg_SC_holds}
The results from \cite{TonegawaWickramasekera10} give
$\dimh B(x_j,r) \cap \sing V \leq n-7$ for every $j=1,\dots,N$.
As the balls $\{B(x_j,r)\}$ cover $U \cap \spt \norm{V}$, the same holds for
$\sing V$.

\ref{item:reg_SC_fails}
When \ref{item:stability_condition_reg} fails,
there must be a subsequence $(u_{i'})$ that is unstable in one of the balls
$B_r$ of the cover, so that in its complement
\begin{equation}
\ind_{U \setminus \overline{B_r}} u_{i'} \leq k-1
\quad \text{for all $i'$.}
\end{equation}
The conclusion follows from the induction hypothesis applied
to $(u_{i'})$ in $U \setminus \overline{B_r}$.
\end{proof}
Now consider a decreasing sequence $r_m \to 0$. For every $m$, choose
points $x_1^m,\dots,x_{N_m}^m \in U \cap \spt \norm{V}$ such that
$U \cap \spt \norm{V} \subset \cup_{j=1}^{N_m} B(x_j^m,r_m)$.
Then either \ref{item:stability_condition_reg} holds for the cover
$\{ B(x_j^m,r_m)\}$ constructed for some $m$,
in which case we can conclude from
Claim~\ref{claim:reg_SC_holds},
or else there is sequence of points $(y_m)$ in $U \cap \spt \norm{V}$ for which
\begin{equation}
\dimh \sing V \setminus \overline{B}(y_m,r_m) \leq n-7.
\end{equation}
Possibly after extracting a subsequence, the sequence $(y_m)$ converges to a point
$y \in \overline{U} \cap \spt \norm{V}$.
As $U \setminus \{y\} \subset
\cup_{m \geq 0} U \setminus \overline{B}(y_m,r_m)$, we get
$\dimh \left(\sing V \setminus \{y\}\right) \leq n-7$.

If $n \geq 7$, then $\dimh \sing V \leq n-7$ holds whether or not $y \in \sing V$,
as points are zero-dimensional.
If however $2 \leq n \leq 6$ then we need $\sing V = \emptyset$, which
amounts to the following claim.

\begin{claim}
	\label{claim:y_not_in_sing_V}
	If $2 \leq n \leq 6$ then $y \notin \sing V$.
\end{claim}

\begin{proof}
Choose $B(y,R) \subset U$, and
consider balls $\{B(y,R_m)\}_{m \in \NN}$ with
shrinking radii $R_m := 2^{-m}R$.
If for some $m$ there is a subsequence $(u_{i'})$ with
\begin{equation}
\ind_{B(y,R_m)} u_{i'} \leq k-1
\quad \text{for all $i'$,}
\end{equation}
then we can conclude from the induction hypothesis.
Otherwise for all $m$
\begin{equation}
\ind_{B(y,R_m)} u_{i} = k
\quad \text{for $i$ large enough,}
\end{equation}
and the $u_i$ are eventually stable in the annulus $B(y,R) \setminus \overline{B}(y,R_m)$.
By Theorem~\ref{thm:main_thm_propagation_of_index_bounds}\ref{item:main_thm_ind_reg}
\begin{equation}
\ind_{B(y,R) \setminus \overline{B}(y,R_m)} B_V = 0
\text{ for all $R_m \to 0$,}
\end{equation}
and thus $\ind_{B(y,R) \setminus \{y\}} B_V = 0$.

By contradiction, suppose that $y \in \sing V$. Then $\ind_{B(y,r)} B_V = 0$
holds in the whole ball $B(y,r)$ away from $\sing V$, and
the regularity results of~\cite{Wickramasekera14} give
$\dimh B(y,R) \cap \sing V \leq n-7$, so that $y \notin \sing V$.
\end{proof}
Claim~\ref{claim:y_not_in_sing_V} concludes the induction step;
together with the base of induction, we have proved that
$\dimh \sing V \leq n-7$. This finishes the proof of
Theorem~\ref{thm:main_thm_propagation_of_index_bounds}.

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text:
Professor Evgeny Oscarovich Paton (,  1870–1953) was a Russian and Soviet engineer of Ukrainian descent who established the E. O. Paton Electric Welding Institute in Kyiv. Paton was a people's deputy of the Supreme Soviet of the Soviet Union (1946–1953). He was the father of Borys Paton.

Early career
Born in 1870 in Nice, France, he studied at the Dresden Technical University (graduated in 1894), and at the Petersburg Institute of Railway Roads (graduated in 1896). He designed the structure of the train station in Dresden, and was a lecturer at the  Moscow Engineering College of Railway Roads (1889–1904). Paton was a professor at Kyiv Polytechnic Institute and the Chairman of the Bridge Department from 1904 to 1938. In 1929 he organized a welding laboratory and the Electric Welding Committee. In 1934 Paton founded the Electric Welding Institute of the Academy of Sciences of Ukrainian SSR in Kyiv. From 1945 to 1952 he was a Vice-President of the Academy of Sciences of Ukrainian SSR.

Welding research
Paton was a pioneer researcher of joining and welding technology. In order to make welding a reliable technological process, it was necessary to conduct comprehensive research regarding the mechanics of welded structures, the metallurgical processes involved, and the physics of the arc, as well as to develop welding equipment, tools, and techniques.

Paton created methods used in the design of rational bridge spans, investigated the conditions of their operation, and suggested methods for restoring damaged bridges. He carried out research on the fundamentals of welding, how to calculate the strength of welded structures, and the mechanization of welding processes. He supervised the development of automatic submerged arc welding. During World War II Paton supervised the design and production of  equipment and technology for automated welding of special steels for tanks, bombs, and other military hardware.

Paton supervised the wide implementation of welding in industry, including the design and production of assembly-line welding systems. He designed welded bridges and founded a domestic school of metal welding. Paton was awarded almost all of the highest Soviet government and scientific awards and prizes, and received the title of the Hero of Socialist Labour.

References

External links
 Biography at the E.O. Paton Electric Welding Institute
 Biography at the Kyiv Polytechnic Institute (in Ukrainian)
 Biography at the Kyiv Polytechnic Institute (in English)

1870 births
1953 deaths
Burials at Baikove Cemetery
People from Nice
French people of Russian descent
History of Kyiv
Bridge engineers
Russian civil engineers
Ukrainian bridge engineers
Ukrainian civil engineers
Heroes of Socialist Labour
Soviet engineers
20th-century Russian engineers
Russian inventors
20th-century Ukrainian engineers
Ukrainian inventors
TU Dresden alumni
Russians in Ukraine
Igor Sikorsky Kyiv Polytechnic Institute
Soviet inventors

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text:
Ought we compel people to be vaccinated?
Posted on Monday, Aug 10, 2020 2:10AM Monday, August 10, 2020 by Gerald Dworkin
by David Copp and Gerald Dworkin
We have argued in a recent article in 3 Quarks Daily that there is a moral obligation for those who are able to safely be vaccinated against serious diseases such as measles and COVID-19 to do so. The gist of the argument is that, when certain factual conditions are met, people have a duty to accept vaccination. There are two conditions. First, there is a significant benefit to the person who is vaccinated and little risk. Second, there is a significant benefit to those to whom the person might otherwise transmit the disease. The benefit is a reduction in the risk of serious harm, which obtains if the disease is life-threatening or seriously threatening of significant pain, physical or emotional damage, or significant expense or effort to avoid these, to a significant subset of the population. We will refer again to these conditions, so let's have a term for them. Call them the "factual preconditions" of an obligation to accept vaccination.
In this essay we are interested in a different set of issues–those surrounding the use of legal coercion to enforce the moral duty above. It is important to see that the two questions are distinct. There are moral obligations that we do not use the law to enforce. For example, we are generally obligated to tell the truth. People often lie when they are obligated not to do so. But lying is not generally a criminal offense. Suppose that someone who is seriously hurt and trying to get to an emergency room in his car, asks us where the nearest emergency room is. If we gave him false directions, we would clearly be failing to comply with a moral obligation to tell the truth in such circumstances. But, to our knowledge there is no law which requires us to tell the truth in such a circumstance.
This, of course, only shows there is no existing legal enforcement of all truth telling–except in many particular cases, e.g. court testimony, taking out a bank loan, etc. Perhaps, as they say, there oughta be a law that requires truth telling. But a little thought shows that there are many good reasons not to have such a law. Such a law would be generally unenforceable. For another thing, such a law would be overly broad, since people are sometimes entitled to speak falsely. Further, people have a right to privacy that would be infringed by an overly broad law requiring truth telling. Most important, we have a right to free speech and a law requiring us to speak truly would infringe this right.
So the existence of a moral obligation to do something is not in general sufficient to justify having a law that requires it. There are cases where violating a moral obligation would harm no-one. I ought to apologize when I do something wrong, but by not apologizing I normally do not harm anyone. No-one thinks there should be a law that requires apologizing. There are cases where there is a moral obligation to do something but we have a right not to do it. In such cases, as with the speech example, it arguably would be illegitimate to create a law that requires these things even if they are obligatory.
Moreover, the existence of a moral obligation to do something is not in general necessary to justify having a law that requires it. Consider traffic laws. There is a moral obligation not to drive recklessly or carelessly. But prior to legislation, there was no moral obligation to drive on the right or left side of a highway, or to drive at this or that speed. Yet it is legitimate to have laws that require us to drive on the right or the left as well as laws that set speed limits. Such laws are for the public good. People can object that such laws interfere with our freedom, and they do, but they are justified by the harm they prevent.
Our argument that there is a moral obligation to get vaccinated was based in the harms that not vaccinating causes or risks causing to others. Our argument that it would be legitimate to have a law that requires vaccination will also draw on these harms and risks of harm. For, in general we think, it is legitimate to have laws requiring people not to cause harm, or risk of harm, to others. Here, what we mean by "harming someone", is to wrongfully set back an interest of theirs that they have a right to have protected. According to this definition, for example, you do not harm me if you beat me in a fair competition, or if you give me a failing grade on an exam, because even though you set back an interest of mine, I have no right to be protected against losing or against failing the exam. But you do harm me if you punch me in the nose, or put a drug in my drink, because by doing these things you set back an interest of mine and you do this despite my having a right to be protected in this interest.
Perhaps there are cases where it would not be legitimate to have a law requiring or prohibiting some harm, even with this definition of harm. These would be cases where there is a right to have an interest protected, but not to have it protected by law. We will discuss some apparent examples. But we will argue that the state can legitimately protect us against the harms and risks created by those who do not accept vaccination — at least in case of a public health crisis, and perhaps more generally — provided that our factual preconditions are satisfied.
Notice that even if it would be legitimate to have a law requiring people to get vaccinated, it does not follow that it would be good public policy to have such a law. And even if it would be a good thing to have such a law, many questions remain about the details, such as questions about how stringent the requirement should be, and about whether it would apply in general or only in public health emergencies. We do not think the law should require people to be forcibly vaccinated. We will argue instead for a law requiring vaccination that penalizes refusal with a fine. So we need to deal with three issues. Call them the legitimacy issue, the public policy issue, and the issue of details.
In the U.S. context, people will often wonder whether a proposed law would be constitutional. We shall set aside this issue and instead focus on background philosophical issues .
There have been laws that require vaccination. Massachusetts enacted a smallpox vaccination requirement in 1809. It was designed to protect public health and safety, and it was held constitutional in Jacobson v Massachusetts. As a condition of school enrollment, state governments require children to be vaccinated against an array of vaccine preventable diseases, including diphtheria, measles, rubella, and polio. To be sure, many states allow exemptions for medical reasons, or for reasons of belief, including religious beliefs. Some states allow vaccination to be required In a public health emergency. In such an emergency, individuals who refuse to be vaccinated for health reasons or reasons of belief might be required to quarantine themselves. Americans generally accept that military conscription is a legitimate use of state power, and the military requires troops to be vaccinated against a range of vaccine preventable diseases.
Our current freedom from the scourge of a variety of diseases that used to kill and harm many people is due to widespread and often mandated vaccination or other public health measures. Smallpox was eradicated in the 1970s due to widespread vaccination. Polio has almost been eradicated due to widespread vaccination. The impact of measles has been greatly reduced due to widespread vaccination. In each case, vaccination requirements have played a role, either requirements imposed on school children as a condition of enrollment or, as in the case of the fight against smallpox in Massachusetts, wider requirements. Those who refuse vaccination against these diseases, and others, are free-loading on the submission of countless others to public health requirements to be vaccinated.
There are also requirements that are analogous to requiring vaccination. Many states have imposed legal requirements to wear face masks when in public, justified by public health concerns during the COVID-19 pandemic. Many if not most jurisdictions require drivers to wear seatbelts and motorcyclists to wear helmets at least partly out of public health concerns.
It is true that vaccination involves a physical intrusion into the body, and there are very few legal requirements to submit to such intrusions as opposed to penalties if one does not. But there are clearly laws which allow physical intrusion. Police can, provided they obtain a search warrant, conduct body cavity searches to search for evidence of a crime such as drug possession. They can conduct body cavity searches at a border or airport without a search warrant. Drivers can be required to submit to a breath test or a blood test if they are suspected of driving while intoxicated. Perhaps the closest analogy to requirements to be vaccinated are the laws that allow forced medication of a psychotic patient to render him competent to be tried for his crime.
Of course, one could argue that all such requirements are illegitimate, so the existence of these laws can only show that a law requiring actual vaccination would not be unprecedented.
Without taking a stand on laws permitting forced vaccination we will be dealing exclusively with laws which require vaccination under penalty of fine or imprisonment.
Our argument rests on the harms and risks of harm created by not getting vaccinated, or by not having one's children vaccinated. In general, we think, a state is justified to prohibit behavior that seriously harms, or creates significant risk of serious harm to others — where, again, by "harming someone" we mean to wrongfully set back an interest of theirs that they have a right to have protected.
Are there exceptions, cases in which a state would not be permitted to prohibit behavior even though it would seriously harm others or create a serious risk of harm to others? Can we be required to donate blood if we have a blood type that is in short supply? Not donating does not harm anyone, it only fails to help them. So the harm principle does not clearly apply. Requiring the donation is not requiring us not to harm others. Can we legitimately be prohibited from driving since in driving we set back the interests of other people who use the roads by creating a risk of injury, property damage, or death? We don't think that it is wrong to create such a risk simply by driving, so we don't think we harm other people simply by driving. But we think we would harm other people by driving in a reckless manner or by driving while intoxicated. And we are legitimately required by law not to drive in these ways. We are required to do what we can to reduce the risk of harming others when we drive. We think it would similarly be legitimate to require people by law to reduce the risk of infecting others by getting vaccinated.
What specifically are the harms and risks of non-vaccination. We think there are five. First, non-vaccinators create a risk that they will transmit the disease to others if they get it. Second they actually do harm these others if they get the disease and transmit it. Third, they are weakening the community's protective herd immunity, or reducing the chance that the community will achieve herd immunity, even if only to a small degree, which increases the risk to everyone that the disease will spread in the population. (Herd immunity exists when enough people in a population have achieved immunity to the disease, either through vaccination or infection, to make its spread unlikely.) Fourth, if herd immunity has been established in the community due to the behavior of others, who have gone to the trouble of getting vaccinated, the non-vaccinators are free-loading. Finally, fifth, if the non-vaccinators have children and refuse to vaccinate them, then they are creating a risk to their children. For if the factual preconditions we mentioned at the outset are true, we are dealing with a disease that is significantly worse than any risk from the vaccine.
In a public health emergency involving a disease like COVID-19, the harms and risks of harm are greater. For herd immunity has not been achieved and the disease is actively spreading in the community. Once there is a vaccine, herd immunity can be achieved, but only if enough people get vaccinated. Not getting vaccinated reduces the chance that the community will achieve herd immunity, which increases the risk to everyone that the disease will continue to spread in the population. Also non-vaccinators are at serious risk of getting the disease and, if they do, there is the risk that they will transmit it to others.
These harms and risks can be somewhat mitigated if non-vaccinators quarantine themselves, so a requirement to accept vaccination could provide the alternative of quarantine. It might require a person either to accept vaccination or to be quarantined.
All in all, then, we think that the harm principle would justify the state in requiring people to accept vaccination provided that the factual preconditions we mentioned earlier are met. Note that a person who cannot safely be vaccinated for some reason must be exempted from the requirement. This is because one of the key preconditions is that the vaccine promises to bring a significant benefit to the person who is vaccinated and little risk.
So much for the legitimacy issue. We now turn to the public policy issue. Granted that it would be legitimate, we still must face the question whether it would be good public policy to require vaccination. The fundamental standard to use, in thinking about this, we think, is a kind of efficiency standard. Would a given proposed policy regarding vaccination be reasonably expected to lead the society in question to achieve, or would it significantly help the society to achieve, or to sustain, herd immunity?
This issue is intertwined with the question of detail, since it might be good public policy to fine people who fail to accept vaccination even if it would not be good public policy to permit public health officials to forcibly vaccinate people. To simplify, we will only consider a policy of fining people who fail to accept vaccination, either for themselves or their minor children. Such a policy might treat a refusal to accept vaccination as a misdemeanor. It would amount to a failure to show reasonable consideration for the health of other people.
There is the issue of cost. We assume that if there is a good reason to have a law requiring vaccination, the cost of the vaccination should be borne by the state or included in insurance. A model for how to do this is the flu vaccine.
Clearly, as we have already said, a well-designed vaccination requirement would exempt people who cannot safely be vaccinated for some medical reason. Should a well-designed law also exempt people who have religious or philosophical objections to vaccines? We think not. People should not be permitted to harm or to create risk of harm to others, not even if they have religious or philosophical objections to avoiding creating the harm or the risk. But since some people have objections of these kinds, the law arguably should permit them instead to make a different kind of contribution to public health, much as conscientious objectors to military service are permitted instead to make a different kind of contribution to the public good.
The issue whether such a policy would be a good one to implement depends on social and political circumstances in the jurisdiction in question. A policy that would be effective in one social environment might be highly ineffective in another one. This is for at least five reasons. The core reason is the problem of non-compliance. There is the ideology of freedom. There are the problems of misinformation and misunderstanding, either about this vaccine or vaccines in general. There is the problem of lack of trust. And there is the problem of enforcement.
Some citizens will not comply with a requirement to be vaccinated. Others will resist strenuously and perhaps even violently. If enough people in a certain jurisdiction will not comply, or will resist, especially if they will resist violently, a vaccination requirement will not likely be successful in getting us to herd immunity. Given the recent experience with requirements to wear face masks in the United States, It is clear that there will be non-compliance and resistance in the U.S. to a requirement to allow oneself to be vaccinated. Police officers who are expected to enforce such a requirement might themselves refuse to comply, and refuse to enforce the requirement, in the way that some refused to comply with face mask requirements.
We have already alluded to one reason why people might refuse to comply and might even resist a vaccination requirement. They might view the requirement as an illegitimate interference with their liberty. A philosophical reply likely wouldn't be an effective way to combat this idea since it likely is held for ideological reasons rather than on the basis of argument. Appeals to a concern for the public good, or to a duty to contribute to the public good might help.
Another reason to expect there would be widespread non-compliance, and even resistance, to a vaccination requirement is the widespread misinformation or at least misunderstanding about vaccines. We refer to the anti-vax movement and its promotion of misinformation about vaccines. There are also conspiracy theories about vaccines. To be effective, a vaccination requirement would need to be accompanied or preceded by an information campaign that explained how vaccines work and how it can be known that they are safe and effective. This campaign would have to target a generalized fear of vaccines as well as worries about particular vaccines.
Lack of trust in government and in particular public health agencies, such as the Centers for Disease Control and the World Health Organization, is yet another obstacle to the effectiveness of a vaccination requirement. Can this particular vaccine be trusted to be safe and effective? Even if people are not worried about vaccines in general, they might worry about a particular vaccine that has been developed and approved in a rush, under political pressure.
Finally, there is the problem of enforcement. We have already mentioned the point that police officers who are expected to enforce the requirement might refuse to do so. They might themselves be moved by the ideology of freedom, by misinformation, or by lack of trust. The problem is deeper than this, we think, since public policy leaders themselves sometimes fail to support public health requirements and might resist enforcing a vaccination requirement. They might also be intimidated by those who would violently resist a vaccination requirement.
Unfortunately all of these obstacles to the effectiveness of a vaccination requirement, and strategies for combating them, would have to be weighed carefully. The paramount goal must be to achieve and sustain herd immunity against vaccine preventable diseases, at least when our factual preconditions are met. There can be no fixed rule about how to do this. These issues are especially pressing now, during a public health emergency, when a dangerous disease is spreading in an uncontrolled way through a vulnerable population. To control the disease we need a vaccine, and we need enough people to accept vaccination. If appeals to health concerns and to public spiritedness are not enough, and if information campaigns are not sufficient to overcome resistance to a vaccine, a vaccination requirement is one tool to consider. Yet it is not an easy tool to use with success.
Fortunately, there might be another way forward. According to research by the Associated Press-NORC Center for Public Affairs, in late May, only about half of Americans said they would get a COVID-19 vaccine if one were available, but only one in five said they would refuse. Another 31% simply weren't sure if they'd get vaccinated. Acceptance by about 70% of the population might be sufficient to achieve herd immunity, so if this data from late May remains accurate, a nudge might be enough to get the population over the line. What kind of nudge? It might be sufficient to issue an order requiring vaccination even if no penalty is attached to a failure to follow the order.
David Copp and Gerald Dworkin are both Distinguished Professors of Philosophy Emeritus at the University of California, Davis.
"Mandatory Vaccinations: Precedent and Current Laws," Kathleen S. Swendiman, Legislative Attorney, March 10, 2011 Congressional Research Service 7-5700 www.crs.gov RS21414
Monday Poem
Mindfulness and Social Identity
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