Nodal geometry, heat diffusion and Brownian motion
Bogdan Georgiev and Mayukh Mukherjee
We use tools from $n$-dimensional Brownian motion in conjunction with the Feynman-Kac formulation of heat diffusion to study nodal geometry on a compact Riemannian manifold $M$. On one hand we extend a theorem of Lieb (see \cite{L) and prove that %on a compact Riemannian manifold $M$ any nodal domain $\Omega_\lambda$ almost fully contains a ball of radius $\sim \frac{1{\sqrt{\lambda$, %corresponding to an eigenvalue $\lambda$ %contains a point $x$ such that the volume of $B_{\frac{C{\sqrt{\lambda(x) \setminus \Omega_\lambda$ can be made smaller than any arbitrary small constant which is made precise by Theorem \ref{th:Large-Inscribed-Ball below. This also gives a slight refinement of a result by Mangoubi, concerning the inradius of nodal domains (\cite{Man2). On the other hand, we also prove that no nodal domain can be contained in a reasonably narrow tubular neighbourhood of unions of finitely many submanifolds inside $M$ (this is Theorem \ref{stein).
Nodal-Geometry-GM.tex
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\begin{document}
\title[Nodal geometry, heat diffusion and Brownian motion]{ Nodal geometry, heat diffusion and Brownian motion}
\author{Bogdan Georgiev and Mayukh Mukherjee}

\address{Max Planck Institute for Mathematics\\ Vivatsgasse 7\\ 53111 Bonn,
\\ Germany}

\email{bogeor@mpim-bonn.mpg.de}
\email{mukherjee@mpim-bonn.mpg.de}
\begin{abstract} 


We use tools from $n$-dimensional Brownian motion in conjunction with the Feynman-Kac formulation of heat diffusion to study nodal geometry on a compact Riemannian manifold $M$. On one hand we extend a theorem of Lieb (see \cite{L}) and prove that %on a compact Riemannian manifold $M$ 
any nodal domain $\Omega_\lambda$ almost fully contains a ball of radius $\sim \frac{1}{\sqrt{\lambda}}$, %corresponding to an eigenvalue $\lambda$ 
%contains a point $x$ such that the volume of $B_{\frac{C}{\sqrt{\lambda}}}(x) \setminus \Omega_\lambda$ can be made smaller than any arbitrary small constant 
which is made precise by Theorem \ref{th:Large-Inscribed-Ball} below. This also gives a slight refinement of a result by Mangoubi, concerning the inradius of nodal domains (\cite{Man2}). On the other hand, we also prove that no nodal domain can be contained in a reasonably narrow tubular neighbourhood of unions of finitely many submanifolds inside $M$ (this is Theorem \ref{stein}). 
\end{abstract}
\maketitle
%\todo{Check Abstract}
\section{Introduction}
We consider a compact $n$-dimensional smooth Riemannian manifold $M$, and the Laplacian (or the Laplace-Beltrami operator) $-\Delta$ on $M$\footnote{We use the analyst's sign convention, namely, $-\Delta$ is positive semidefinite.}. For an eigenvalue $\lambda$ of $-\Delta$ and a corresponding eigenfunction $\varphi_\lambda$, %Our setting: let $M$ be a compact Riemannian manifold with Laplacian $-\Delta$, whose eigenvalues (with multiplicity) are $0 \leq \lambda_1 \leq \lambda_2 \leq .....$, and (smooth) eigenfunctions $\varphi_n$. In this short note, we make some (quantitative) observations about the shape of the  nodal domain of eigenfunctions. R
recall that a nodal domain $ \Omega_\lambda $ is a connected component of the complement of the nodal set $N_{\varphi_\lambda} := \{ x \in M : \varphi_\lambda (x) = 0\}$. In this paper, we are interested in the asymptotic geometry of a nodal domain $\Omega_\lambda$ as $ \lambda \rightarrow \infty $.

In particular, we address the following two questions.

First, we start by discussing the problem of whether a nodal domain can be squeezed in a tubular neighbourhood around a certain subset $ \Sigma \subseteq M $. A result of Steinerberger (see Theorem 2 of ~\cite{St}) states that for some constant $ r_0>0 $ a nodal domain $ \Omega_\lambda $ cannot be contained in a $ \frac{r_0}{\sqrt{\lambda}} $-tubular neighbourhood of hypersurface $ \Sigma $, provided that $ \Sigma $ is sufficiently flat in the following sense: $ \Sigma $ must admit a unique metric projection in a wavelength (i.e. $\sim \frac{1}{\sqrt{\lambda}} $) tubular neighbourhood.
The proof involves the study of a heat process associated to the nodal domain, where one also uses estimates for Brownian motion and the Feynman-Kac formula.

We relax the conditions imposed on $ \Sigma $. A direct extension is

\begin{proposition}\label{babycase}
	There is a constant $r_0$ depending only on $(M, g)$ such that if a submanifold $\Sigma_\lambda \subset M$ is admissible up to distance $\frac{1}{\sqrt{\lambda}}$, then no nodal domain $\Omega_\lambda$ can be contained in a $\frac{r_0}{\sqrt{\lambda}}$-tubular neighbourhood of $\Sigma_\lambda$.
\end{proposition}

For the definition of admissibility, see Definition \ref{adm} below. Roughly, Proposition \ref{babycase} allows $ \Sigma_\lambda $ to not necessarily be a hypersurface, i.e. it is allowed to have an arbitrary dimension $ 1 \leq k \leq n-1 $. Further, it turns out that we can select $ \Sigma_\lambda $ to be a union of submanifolds of varying dimensions, having relaxed admissibility conditions (see Definition \ref{def:Alpha-Admissibility}). We have,

\begin{theorem}\label{stein}
	Given an $ \alpha $-admissible union $\Sigma_\lambda$, there exists a constant $C$, independent of $\lambda$, such that $N_{\frac{C}{\sqrt{\lambda}}}(\Sigma_\lambda)$ cannot fully contain a nodal domain $\Omega_\lambda$.
\end{theorem}

%The strategy is to apply refined Brownian motion estimates. We note that 
Theorem \ref{stein} gives a indication as to the ``thickness'' or general shape of a nodal domain in many situations of practical interest. For example, in dimension $2$, numerics show nodal domains to look like a tubular neighbourhood of a tree. We also note that our proof of Theorem \ref{stein} reveals a bit more information, but for aesthetic reasons, we prefer to state the theorem this way.
Heuristically, the proof reveals that the nodal domain $\Omega_\lambda$ is thicker at the points where the eigenfunction $\varphi_\lambda$ attains its maximum, or at points where $\varphi_\lambda (x) \geq \beta\text{max}_{y \in \Omega_\lambda}|\varphi_\lambda (y)|$, for a fixed constant $\beta > 0$. 

Second, we study the problem of how large a ball one may inscribe in a nodal domain $ \Omega_\lambda $ at a point where the eigenfunction achieves extremal values on $ \Omega_\lambda $. We show

\begin{theorem} \label{th:Large-Inscribed-Ball}
Let $\dim M \geq 3, \epsilon_0 > 0 $ be fixed and $ x_0 \in \Omega_\lambda $ be such that $ |\varphi_\lambda(x_0)| = max_{\Omega_\lambda}|\varphi_\lambda| $. There exists $ r_0 = r_0 (\epsilon_0) $, such that
	\begin{equation}\label{Vol}
		\frac{\Vol\left( B_{r_0} \cap \Omega_\lambda \right)}{\Vol\left(B_{r_0}\right)} \geq 1 - \epsilon_0,
	\end{equation}
	where $ B_{r_0} $ denotes $ B\left(x_0, \frac{r_0}{\sqrt{\lambda}}\right) $.
\end{theorem}

%despite the fact that the tools are more or less well-known.

%\todo{Check Paragraph}
A celebrated theorem of Lieb (see \cite{L}) considers the case of a domain $ \Omega \subset \mathbb{R}^n $ and states that there exists a point $ x_0 \in \Omega $, where a ball of radius $ \frac{C}{\sqrt{\lambda_1(\Omega)}} $ can almost be inscribed (in the sense of our Theorem \ref{th:Large-Inscribed-Ball}). A further generalization was obtained in the paper ~\cite{MS} (see, in particular, Theorem 1.1 and Subsection 5.1 of ~\cite{MS}). However, the point $ x_0 $ was not specified. Physically, one expects that $ x_0 $ is close to the point where the first Dirichlet eigenfunction of $ \Omega $ attains extremal values. This is in fact the essential statement of Theorem \ref{th:Large-Inscribed-Ball} above. %As regards the statement of Theorem \ref{th:Large-Inscribed-Ball}, 
Also, in this context, it is illuminating to compare the main Theorem from ~\cite{CD}.

As a corollary of Theorem \ref{th:Large-Inscribed-Ball}, we derive the following:
\begin{corollary} \label{cor:Large-Ball}
	Let $ \Omega_\lambda $ be a nodal domain upon which the corresponding eigenfunction $ \varphi_\lambda $ is positive. Let $ x_0 $ be a point of maximum of $ \varphi_\lambda $ on $ \Omega_\lambda $. Then there exists a ball $ B\left(x_0, \frac{C}{\lambda^{\alpha(n)}}\right) \subseteq \Omega_\lambda $ with $ \alpha (n) = \frac{1}{4}(n-1)+\frac{1}{2n} $ and a constant $ C = C(M, g) $.
\end{corollary}
This recovers Theorem 1.5 of ~\cite{Man2}, with the additional information that the ball of radius $\frac{C}{\lambda^{\alpha(n)}}$ is centered around the max point of the eigenfunction $\varphi_\lambda$ (for more discussion on this, see Section \ref{LBMP}).

%In this paper, our heat theoretic tools allow us to approach the inradius estimate of $\text{inrad }(\Omega_\lambda)$ in a way that is different from ~\cite{Man1} and ~\cite{Man2}, and also the aforementioned references. A main reference for us is ~\cite{St}, which cites ~\cite{GJ} as a precursor to many of these ideas. We also draw on ideas and motivation from ~\cite{GS}, as also previous independent work by Saloff-Coste and Grigor'yan. 

\subsection{Basic heuristics}
We outline the main idea behind Proposition \ref{babycase} and Theorems \ref{stein} and \ref{th:Large-Inscribed-Ball}.

First, one considers a point $ x_0 \in \Omega_\lambda$ where the eigenfunction achieves a maximum on the nodal domain (w.l.o.g. we assume that the eigenfunction is positive on $ \Omega_\lambda $). One then considers the quantity $ p(t,x_0) $ - i.e. the probability that a Brownian motion started at $ x_0 $ escapes the nodal domain within time $ t $.

The main strategy is to obtain two-sided bounds for $ p(t, x_0) $.

On one hand, we have the Feynman-Kac formula (cf. Appendix $ 2 $) which provides a straightforward upper bound only in terms of $ t $ (see Equation (\ref{stein1}) below).

On the other hand, depending on the context of the theorems above, we provide a lower bound for $ p(t,x_0) $ in terms of some geometric data. To this end, we take advantage of various tools some of which are: formulas for hitting probabilities of spheres and the parabolic scaling between the space and time variables; comparability of Brownian motions on manifolds with similar geometry (cf. Appendix $ 2 $); bounds for hitting probabilities in terms of $ 2 $-capacity (cf. \cite{GS}), etc.

%Indeed, the main idea of the paper is to use the parabolic scaling inherent to the Brownian process effectively. To wit, 
%The following observation is crucial for our results: consider a Brownian particle in $\RR^n$ starting at the origin, and a sphere $S^{n - 1}_r$ of radius $r$ centered at the origin. If $t$ is the time taken by the particle to hit $S^{n - 1}_r$, then it is clear that increasing $t$ and increasing $r^2$ will offset each other's effect, as per the parabolic scaling (see equation (\ref{JKF}) below). In this paper, we take advantage of the ability to vary $t$ and $r$ independently of each other, that is, we make use of  the parabolic scaling inherent to the Brownian process effectively. Roughly, the idea behind Theorem \ref{th:Large-Inscribed-Ball} is that %one can use Brownian motion and capacity estimates to prove that 
%a Brownian particle starting at the center of a ball of radius $r$ will escape the ball with high probability by time $t \approx \sqrt{r}$. However, if the center of the ball is a maximum point of an eigenfunction, the Brownian particle will have a low probability of moving out of the nodal domain $\Omega_\lambda$ by time $\approx \sqrt{r}$ if $r$ is small (see (\ref{stein1}) below). Taken together, these tell us (via tools developed in ~\cite{GS}) that the $2$-capacity of the ball outside $\Omega_\lambda$ is negligible, which can be translated into (\ref{Vol}) via a typical isocapacitary inequality. %in combination with an upper bound, derived in ~\cite{St}, on a heat diffusion process associated to the nodal domain. 

 %respecting the parabolic scaling. 

%\todo{Better explanation maybe}
%Our results do not of course seem optimal, but we believe that our methods appear to be natural when one discusses the inradius of a nodal domain. Roughly, because it is expected that the inradius should be wavelength (i.e. $\frac{1}{\sqrt{\lambda}} $) and the parabolic scaling of the heat equation is a feature suiting this heuristic.

%provide insights into, for example, why the optimal inradius estimate of a nodal domain $\Omega_\lambda$ is expected to be $\sim \frac{1}{\sqrt{\lambda}}$ and not, say, $\sim \frac{1}{\lambda}$; this is again intrinsically tied to the parabolic scaling.

%We note that Theorem \ref{th:Large-Inscribed-Ball} is not entirely new in the literature; there is the well-known result of Lieb, see Corollary 2 of ~\cite{L}. However, Lieb's proof was based on variational arguments, making our approach very different from his. Moreover, the fact that the ``almost optimal'' ball $B_{r_0}$ can be placed around the maximum point $x_0$ is an additional insight which follows naturally in our set up. 
\subsection{Outline of the paper}
In Section \ref{Int}, we recall tools from $n$-dimensional Brownian motion and the Feynman-Kac formulation of heat diffusion, and discuss the parabolic scaling technique we referred to above. %recall the tools from Bessel processes and Feynman-Kac heat diffusion and make a simple extension of a result of Steinerberger (see Theorem 2 of ~\cite{St}) heuristically stating the following: given a ``sufficiently flat'' hypersurface $\Sigma \subset M$ there exists a constant $c$ independent of $\lambda$ such that no $\frac{c}{\sqrt{\lambda}}$ tubular neighbourhood of $\Sigma$ can fully contain a nodal domain. In Theorem \ref{babycase} below, we extend this to the case where $\Sigma$ can have any dimension, not necessarily a hypersurface. 



In Section \ref{Perf}, we begin by proving Proposition \ref{babycase}. %, which extends Theorem 2 of ~\cite{St} to the case of an ``admissible'' surface of arbitrary dimension. 
We then take the generalization % extending %take the extension of Theorem \ref{babycase} 
 one step further, by considering intersecting surfaces of different dimensions. Our main result in this direction is Theorem \ref{stein}, which gives a quantitative lower bound on how ``thin'' or ``narrow'' a nodal domain can be. %Observe that Theorem \ref{stein} gives a strong indication as to the ``fatness'' or general shape of a nodal domain. For example, in dimension $2$, numerics show nodal domains to look like a tubular neighbourhood of a tree.

In Section \ref{LBMP}, we take up the investigation of inradius estimates of $\Omega_\lambda$. As mentioned before, our main result in this direction is Theorem \ref{th:Large-Inscribed-Ball}. We also establish Corollary \ref{cor:Large-Ball}.

In Appendix 1, Section \ref{sec:Heat-Kernel-Comment} we briefly mention that the admissibility question for $ \Sigma $ having conical singularities may be tackled via an explicit heat kernel expression, and give references to some of the relevant literature. 

Lastly, in Appendix 2, we include some background material on stochastic analysis on Riemannian manifolds, some of which (to our knowledge) is not widely known, but is important to our investigation. We also believe such results to be of independent interest to the community. Of particular mention is Theorem \ref{compare}, which roughly says that if the metric is perturbed slightly, hitting probabilities of compact sets by Brownian particles are also perturbed slightly. This allows us to apply Brownian motion formulae from $\RR^n$ to compact manifolds, on small distance and time scales. 
%Our first object of investigation is the inradius of the nodal domain $\Omega_\lambda$, which is defined as the radius of the largest 


%Another concern 

\subsection{Acknowledgements} It is a pleasure to thank Stefan Steinerberger for his detailed comments on a draft version of this paper, as well as Dan Mangoubi for his comments and remarks. The authors further thank Yuval Peres and Itai Benjamini for advice regarding Martin capacity. The authors also gratefully acknowledge the Max Planck Institute for Mathematics, Bonn for providing ideal working conditions.

\section{Preliminaries: heat equation, Feynman-Kac and Bessel processes}\label{Int}



%The use of heat equation theoretic tools in the study of nodal geometry was initiated by Steinerberger in ~\cite{St}.
Using the probabilistic formulation of the heat equation for the study of nodal geometry, we are largely inspired by the methods in ~\cite{St}. Of course, such ideas have appeared in the literature before; for example, they are implicit in ~\cite{GJ}. Here we extend some ideas of Steinerberger with the help of tools from $n$-dimensional Brownian motion.

Given an open subset $V \subset M$, consider the solution $p_t(x)$ to the following diffusion process:
\begin{align*}
	(\pa_t - \Delta)p_t(x) & = 0, \text{    } x \in V\\
	p_t(x) & = 1, \text{    } x \in \pa V\\
	p_0(x) & = 0, \text{    } x \in V.
\end{align*}

%\todo{Haven't added the 0, 1 reversal thing, seems not necessary}
%$ r_0 $ is arbitrary small, henc
By the Feynman-Kac formula (see Appendix $2$, Subsection \ref{F-K}), this diffusion process can be understood as the probability that a Brownian motion particle started in $x$ will hit the boundary within time $t$. Now, fix an eigenfunction $\varphi$ (corresponding to the eigenvalue $\lambda$) and a nodal domain $\Omega$, so that $\varphi > 0$ on $\Omega$ without loss of generality. Setting $\Phi (t, x) = e^{-\lambda t}\varphi (x)$, we see that $\Phi$ solves 
\begin{align}\label{HF}
(\pa_t - \Delta)\Phi (t, x) & = 0, x \in \Omega\nonumber\\
\Phi (t, x) & = 0, \text{  on   } \{\varphi = 0\}\\
\Phi (0, x) & = \varphi (x), \text{    } x \in V.\nonumber
\end{align}

%Now we rewrite the above described Dirichlet problem in the probabilistic language. Given an open domain $U, f \in L^2(U), x \in U$ and $t > 0$, we have,
Using the Feynman-Kac formula given by Theorem \ref{F-K-T}, we have,
\beq
e^{t\Delta_V}f(x) = \mathbb{E}_x(f(\omega(t))\phi_{U}(\omega, t)), t > 0. 
\eeq
%where $\omega (t)$ denotes an element of the probability space of Brownian motions starting at $x$, and 
%$$
%\phi_U(\omega, t) = 
%\begin{cases}
%1, & \text{if } \omega ([0, t]) \subset U\\
%0, & \text{otherwise. }
%\end{cases}
%$$

%\todo{Different names for $x, x_0 $}
Now, consider a nodal domain $\Omega$ corresponding to the eigenfunction $\varphi$, and consider the heat flow (\ref{HF}). Let $x_0 \in \Omega$ such that $\varphi (x_0) = \V \varphi\V_{L^\infty(\Omega)}$. We use the following upper bound derived in ~\cite{St}:
\begin{align}\label{stein1}
\Phi (x) & = e^{-\lambda t}\varphi (x) = \mathbb{E}_x(\varphi(\omega(t))\phi_{\Omega}(\omega, t))\\
& \leq \V \varphi\V_{L^\infty(\Omega)}\mathbb{E}_x(\phi_{\Omega}(\omega, t)) = \V \varphi\V_{L^\infty(\Omega)}(1 - p_t(x))\nonumber.
\end{align}
Setting $t = \lambda^{-1}$, we see that the probability of the Brownian motion starting at an extremal point $x_0$ leaving $\Omega$ within time $\lambda^{-1}$ is $\leq 1 - e^{-1}$. A rough interpretation is that maximal points $ x $ are situated deeply into the nodal domain.

For ease of presentation, we adopt the following notation: let a compact subset $K$ of $M$ be given, let $\psi_K(t, x)$ denote the probability that a Brownian motion starting at the point $x$ will reach $K$ within time $t$. Using this notation, the last derived upper estimate translates to $\psi_{M \setminus \Omega}(\lambda^{-1}, x) \leq 1 - e^{-1}$.

Now, we consider an $m$-dimensional Brownian motion of a particle starting at the origin in $\RR^m$, %(also called Bessel processes), 
and calculate the probability of the particle hitting a sphere $\{x \in \RR^m : \V x\V \leq r\}$ of radius $r$ within time $t$. By a well known formula first derived in ~\cite{Ke}, we see that such a probability is given as follows:
\beq\label{JKF}
\mathbb{P}(\sup_{0 \leq s \leq t}\V B(s)\V \geq r) = 1 - \frac{1}{2^{\nu - 1}\Gamma (\nu + 1)}\sum^\infty_{k = 1}\frac{j_{\nu, k}^{\nu - 1}}{J_{\nu + 1}(j_{\nu, k})}e^{-\frac{j^2_{\nu, k}\text{  t}}{2r^2}}, \quad \nu > -1,
\eeq
where $\nu = \frac{m - 2}{2}$ is the ``order'' of the Bessel process, $J_\nu$ is the Bessel function of the first kind of order $\nu$, and $0 < j_{\nu, 1} < j_{\nu, 2} < .....$ is the sequence of positive zeros of $J_\nu$.

Choose $x = x_0, t = \lambda^{-1}$, as before, and let $r = C^{1/2}\lambda^{-1/2}$, where $C$ is a constant to be chosen later, independently of $\lambda$. Plugging this in (\ref{JKF}) then reads,


\beq\label{JKF1}
\mathbb{P}(\sup_{0 \leq s \leq \lambda^{-1}}\V B(s)\V \geq C\lambda^{-1/2}) = 1 - \frac{1}{2^{\nu - 1}\Gamma (\nu + 1)}\sum^\infty_{k = 1}\frac{j_{\nu, k}^{\nu - 1}}{J_{\nu + 1}(j_{\nu, k})}e^{-\frac{j^2_{\nu, k}}{2C}}, \quad \nu > -1.
\eeq 

We need to make a few comments about the asymptotic behaviour of $j_{\nu, k}$ here. For notational convenience, we write $\alpha_k \sim \beta_k$, as $k \to \infty$ if we have $\alpha_k/\beta_k \to 1$ as $k \to \infty$. ~\cite{Wa} gives the asymptotic expansion 
\beq\label{asy1}
j_{\nu, k} = (k + \nu/2 + 1/4)\pi + o(1) \text{   as  } k \to \infty,
\eeq
which tells us that $j_{\nu, k} \sim k\pi$. Also, from ~\cite{Wa}, pp 505, we have that 
\beq\label{asy2}
J_{\nu + 1}(j_{\nu, k}) \sim (-1)^{k - 1}\frac{\sqrt{2}}{\pi}\frac{1}{\sqrt{k}}.
\eeq

These asymptotic estimates, in conjunction with (\ref{JKF1}), tell us that keeping $\nu$ bounded, and given a small $\eta > 0$, one can choose the constant $C$ small enough (depending on $\eta$) such that 
\beq\label{fund}
\mathbb{P}(\sup_{0 \leq s \leq \lambda^{-1}}\V B(s)\V \geq C\lambda^{-1/2}) > 1 - \eta.\eeq

This estimate plays a role in Section \ref{Perf}. In this context, see also Proposition 5.1.4 of ~\cite{Hs}. 

\section{Admissibility conditions and intersecting surfaces}\label{Perf}

We start with the following definition:

%\todo{Is the definition sufficient? Are all of the following claims legit?}
\begin{definition}\label{adm}
	Let $\Sigma_\lambda$ be a smooth family of $k$-dimensional submanifolds (without boundary) embedded in $M$. Call $\Sigma_\lambda$ admissible up to a distance $r$ if $\text{dist} (x, \Sigma_\lambda) \leq r$ implies that there are unique $\Sigma_\lambda^x \in \Sigma_\lambda $ and $y \in \Sigma_\lambda^x$ realizing $\text{dist}(x, \Sigma_\lambda)$, that is, $\text{dist} (x, y) = \text{dist}(x, \Sigma_\lambda)$.
\end{definition}

We note that if $\Sigma_\lambda$ consists of one submanifold which is admissible up to distance $r$, then the above definition means that $r$ is smaller than the normal injectivity radius of $\Sigma_\lambda$. Moreover, the definition implies that $\Sigma_\lambda$ consists of disjoint submanifolds, the distance between every two of which is greater than $r$.

Let us also remark that, Theorem 2 of ~\cite{St} holds true when the hypersurface $\Sigma$ is allowed to vary with respect to $\lambda$ in a controlled way, which is made precise by Definition \ref{adm}. With that clarification in place, Proposition \ref{babycase} is an extension of Theorem 2 of ~\cite{St}.

\begin{proof}[Proof of Proposition \ref{babycase}]
If $\varphi_\lambda$ attains its maximum within $\Omega_\lambda$ at $x_0$, we already know from (\ref{stein1}) that 
\beq\label{eq:Heat-Content-Upper-Bound}
\psi_{M \setminus \Omega_\lambda}(\frac{t_0}{\lambda}, x_0) \leq 1 - e^{-t_0}.
\eeq 

By the admissibility condition on $\Sigma_\lambda$ we know that $x_0$ has a unique metric projection on one and only one $\Sigma_\lambda^{x_0}$ from the collection $\Sigma_\lambda$.


Now, suppose the result is not true. Choose $R, t_0$ small such that Theorem \ref{compare} applies. Choosing $r_0$ sufficiently smaller than $R$, we can find a $\lambda$ such that $\Omega_\lambda$ is contained in a $\frac{r_0}{\sqrt{\lambda}}$-tubular neighbourhood of $\Sigma_\lambda$, denoted by $N_{r_0}^{'} (\Sigma_\lambda)$. %where $r_0$ could be taken arbitrary small (independently of $\lambda$). 
From the remarks after Definition \ref{adm}, it follows that $\Omega_\lambda \subseteq N_{r_0} (\Sigma_\lambda^{x_0})$.

We start a Brownian motion at $x_0$ and, roughly speaking, we see that locally the particle has freedom to wander in $n - k$ ``bad directions'', namely the directions normal to $\Sigma_\lambda^{x_0}$, before it hits $\pa \Omega_\lambda$. That means, we may consider a $(n - k)$-dimensional Brownian motion $B(t)$ starting at $x_0$; see the following diagram:

\includegraphics{Babycase}\label{diag1}
\\
\\

More formally, we choose a normal coordinate chart $(U, \phi)$ around $x_0$ adapted to $\Sigma_\lambda^{x_0}$, where the metric is comparable to the Euclidean metric. We have that $\phi(\Sigma_\lambda^{x_0}) = \phi(U) \cap \{ \mathbb{R}^k \times \{ 0 \}^{n-k} \} $ and $\phi(N_{r_0}(\Sigma_\lambda^{x_0})) = \phi(U) \cap \{ \mathbb{R}^{k} \times [-\frac{r_0}{\sqrt{\lambda}}, \frac{r_0}{\sqrt{\lambda}}]^{n - k} \}$. 
%\todo{Comparability Lemma} 
We take a geodesic ball $B \subset U \subset M$ at $x_0$ of radius $\frac{R}{\sqrt{\lambda}}$. Using the hitting probability notation from Section \ref{Int} and monotonicity with respect to set inclusion we have
\begin{equation}\label{12}
	\psi_{M \setminus \Omega_\lambda}\left(\frac{t_0}{\lambda}, x_0\right) \geq \psi_{B \setminus \Omega_\lambda} \left(\frac{t_0}{\lambda}, x_0\right) \geq \psi_{B \setminus N_{r_0}(\Sigma_\lambda^{x_0})} \left(\frac{t_0}{\lambda}, x_0\right),
\end{equation}
and the comparability result Theorem \ref{compare} implies that, if $c = \frac{t_0}{R^2}$, then there exists a constant $C$, depending on $c$ and $M$, such that 

\begin{equation}
	\psi_{B \setminus N_{r_0}(\Sigma_\lambda^{x_0})}\left(\frac{t_0}{\lambda}, x_0\right) \geq C \psi^e_{\phi(B \setminus N_{r_0}(\Sigma_\lambda^{x_0}))} \left(\frac{t_0}{\lambda}, \phi(x_0)\right),
\end{equation}
%\todo{Comparability Lemma}
where $\psi^e$ denotes the hitting probability in Euclidean space. %and the constant $C$ is given by the comparability lemma. For large enough $\lambda$, one can take $C$ arbitrary close to $1$. Moreover, since the metric is comparable to the Euclidean one, one can find a Euclidean ball $B^e \subseteq \phi(B)$ at $\phi(x_0)$ of radius $\frac{1}{4\sqrt{\lambda}}$. 
We denote $N_{r_0}^e := \phi(N_{r_0}(\Sigma_\lambda^{x_0}))$.

Let us consider the ``solid cylinder'' $S = B_\frac{R}{\sqrt{\lambda}} \times B_\frac{r_0}{\sqrt{\lambda}}$, a product of $k$ and $n-k$ dimensional Euclidean balls centered at $\phi(x_0)$. $S$ is clearly the largest cylinder contained in $ N^e_{r_0} \cap B$. We denote $S = B_1 \times B_2$ for convenience. %Observe that $\psi^e_{\phi(B \setminus N^e_{r_0}(\Sigma_\lambda^{x_0}))} \left(\frac{t_0}{\lambda}, \phi(x_0)\right) \geq  \psi^e_{\pa N^e_{r_0}} \left(\frac{t_0}{\lambda}, \phi(x_0)\right)$. 
By monotonicity,

%\begin{align}\label{endless}
%	 \psi^e_{\phi(B \setminus N_{r_0}(\Sigma_\lambda^{x_0}))} \left(\frac{t_0}{\lambda}, \phi(x_0)\right) + \psi^e_{\pa B_1 \times B_2 }\left(\frac{t_0}{\lambda}, \phi(x_0)\right)  & \geq \psi^e_{\pa N_{r_0}^e}\left(\frac{t_0}{\lambda}, \phi(x_0)\right)\\
%	 & \geq \psi^e_{B_1 \times \pa B_2 }\left(\frac{t_0}{\lambda}, \phi(x_0)\right)\nonumber.
%\end{align}

%\todo{Brownian motion projection}
%Since $r_0$ is sufficiently small compared to $r$, we can say from (\ref{endless}) that %\todo{Equation (15) still makes me suspect that we do not need the small cap comparability. It is trivial by monotonicity anyway. (16) still goes through, it seems.}  
\beq
 \psi^e_{\phi(B \setminus N_{r_0}(\Sigma_\lambda^{x_0}))} \left(\frac{t_0}{\lambda}, \phi(x_0)\right) \geq  \psi^e_{B_1 \times \pa B_2 }\left(\frac{t_0}{\lambda}, \phi(x_0)\right).
\eeq

If $B(t) = (B_1(t),..., B_n(t))$ is an $n$-dimensional Brownian motion, the components $B_i(t)$'s are independent Brownian motions (see, for example, Chapter 2 of ~\cite{MP}). Denoting by $B_k(t)$ and $B_{n - k}(t)$ the projections of $B(t)$ onto the first $k$ and last $n - k$ components respectively, it follows that
\begin{align*}
	\psi^e_{B_1 \times \pa B_2 }\left(\frac{t_0}{\lambda}, \phi(x_0)\right) & \geq \mathbb{P}(\sup_{0 \leq s \leq t_0\lambda^{-1}} \V B_{k}(t)\V \leq \frac{R}{\sqrt{\lambda}}).\mathbb{P}(\sup_{0 \leq s \leq t_0\lambda^{-1}} \V B_{n-k}(t)\V \geq \frac{r_0}{\sqrt{\lambda}})\\
	& \geq c_k \mathbb{P}(\sup_{0 \leq s \leq t_0\lambda^{-1}} \V B_{n-k}(t)\V \geq \frac{r_0}{\sqrt{\lambda}}),
\end{align*}
where $c_k$ is a constant depending on $k$ and the ratio $t_0/R^2$, and can be calculated explicitly from (\ref{JKF1}).
%where $B_{n-k}$ denotes an $n-k$ dimensional Brownian motion.

Using the estimate in Section \ref{Int}, we may take $r_0 \leq R$ sufficiently small so that
\beq\label{16}
	\mathbb{P}(\sup_{0 \leq s \leq t_0\lambda^{-1}} \V B_{n-k}(t)\V \geq 	\frac{r_0}{\sqrt{\lambda}}) > 1 - \varepsilon,
\eeq
where  $\varepsilon$ is sufficiently small. Keeping $c = \frac{t_0}{R^2}$ and (hence) $C$ fixed, we take $t_0$ small enough and $r_0 \leq R$ appropriately, so that (\ref{16}) contradicts (\ref{12}) and the fact that $\psi_{M \setminus \Omega_\lambda}(t_0\lambda^{-1}, x) \leq 1 - e^{-t_0}$.

\end{proof}
\begin{remark}
Note that the constant $r_0$ above is independent of $\Sigma_\lambda$; in other words, the same constant $r_0$ will work for Proposition \ref{babycase} as long as the surface is admissible up to a wavelength distance. Indeed, this results from the fact that $r_0$ depends only on the diffusion process associated to the Brownian motion, and is an inherent property of the manifold itself. 
\end{remark}

Now we discuss generalizations of Proposition \ref{babycase} for $\Sigma_\lambda$ which are more complicated than just a hypersurface, and without the restriction of unique projection. We provide another notion of the admissibility conditions and state a version of Proposition \ref{babycase} by considering situations where $\Sigma_\lambda$ is an union of intersecting surfaces of different dimensions. For a fixed integer $m$, consider $m$ surfaces $\Sigma^i_\lambda, i = 1, 2, .., m$ inside $M$, where $\text{dim }\Sigma^i_\lambda = k_i$ and the surfaces $\Sigma^i_\lambda$ are compact and $C^1$, and can intersect each other non-trivially. The following observation is crucial:
\begin{remark}
Getting rid of the admissibility condition, as in Definition \ref{adm} allows situations where $\Sigma^i_\lambda$ is dense in $M$, for example, $M = \mathbb{T}^2$ and $\Sigma^1_\lambda$ being a generic geodesic. By assuming $\Sigma^i_\lambda$ is compact, we avoid such situations. Also, since we are considering unions of surfaces, the restriction of ``unique projection'' of nearby points, as in Definition \ref{adm}, makes no sense any more, and one can see that the approach of the proof of Proposition \ref{babycase} above does not work. Hence, modifications are needed. 
\end{remark}%satisfy one of the following properties:\newline
%(P1) Each $\Sigma_i$ smooth and satisfies the admissibility condition in Definition \ref{adm}, or\newline
%(P2) Each $\Sigma_i$ is Lipschitz and is compactly embedded inside $M$.

%\begin{definition}\label{def:adm-union}
%	$ \Sigma $ is called an \textit{admissible union}, provided...
%\end{definition}

Call $\Sigma_\lambda = \cup^m_{i = 1} \Sigma^i_\lambda$. We wish to investigate whether an $\varepsilon$-tubular neighbourhood of $\Sigma_\lambda$, which we denote by $N_\varepsilon(\Sigma_\lambda)$, can fully contain a nodal domain $\Omega_\lambda$ of an eigenfunction $\varphi$, where $\varepsilon \sim \frac{1}{\sqrt{\lambda}}$. More formally, let $N_\varepsilon(\Sigma^i_\lambda) = \{x \in M : \text{dist }(x, \Sigma^i_\lambda) < \varepsilon\}$, and let $N_\varepsilon(\Sigma_\lambda) = \cup^m_{i = 1}N_\varepsilon(\Sigma^i_\lambda)$.


%If $\varphi$ attains its maximum within $\Omega_\lambda$ at $x_0$, then by an application of Feynman-Kac, as discussed before

%\begin{equation}\label{eq:Heat-Content-Upper-Bound}
%	\psi_{M \setminus \Omega_\lambda}(C\lambda^{-1}, x_0) \leq 1 - e^{-C}.
%\end{equation}

Monotonicity of the hitting probability function $\psi_K(.,.)$ with respect to set inclusion in $K$ yields
\begin{align}\label{t0r0}
	\psi_{M \setminus \Omega_\lambda}(t, x_0)
	& \geq \psi_{B_{\frac{r_0}{\sqrt{\lambda}}} \setminus \Omega_\lambda }(t,x_0) \\ \nonumber
	& \geq \psi_{B_{\frac{r_0}{\sqrt{\lambda}}} \setminus N_{\frac{r_0}{\sqrt{\lambda}}}(\Sigma_\lambda)}(t, x_0) \\ \nonumber
	& \geq C_\lambda \psi_{\partial B_{\frac{r_0}{\sqrt{\lambda}}}}(t, x_0),
\end{align}
where we introduce $C_\lambda$ - a constant depending only on the family $ \Sigma_\lambda $ and the geometry of $M$, provided the radius $ \frac{r_0}{\sqrt{\lambda}} $ is sufficiently small. We are interested in those $\Sigma_\lambda$ for which $C_\lambda$ is not too small:

\begin{definition} \label{def:Alpha-Admissibility}
	Let $ \alpha > 0 $ be a constant. A union $ \Sigma_\lambda $ as above is called $ \alpha $-admissible, if $ C_\lambda \geq \alpha$.
\end{definition}

As before, our definition of the $ \alpha $-admissibility condition imposes on the union of submanifolds $ \Sigma_\lambda $ a kind of controlled variation in terms of $ \lambda $. In particular, if $ \Sigma_\lambda = \Sigma $ for some fixed union of submanifolds $ \Sigma $, it is $ \alpha $-admissible.

\includegraphics{7}\label{diag2}


%\todo{Picture of a real eigenfunction in $ R^2 $}
We now prove Theorem \ref{stein}. %state our result:
%\begin{theorem}\label{stein}
%Given $\Sigma$ as above, there exists a constant $C$, independent of $\lambda$, such that $N_{\frac{C}{\sqrt{\lambda}}}(\Sigma)$ cannot fully contain a nodal domain $\Omega_\lambda$.
%\end{theorem}
\begin{proof}[Proof of Theorem \ref{stein}]

If the statement is not true, then we may select arbitrarily small $ r_0 $ and find an eigenvalue $\lambda$ with a corresponding inscribed nodal domain $\Omega_\lambda \subset N_{\frac{r_0}{\sqrt{\lambda}}}(\Sigma_\lambda)$.

Choose a point $ x_0 \in \Omega_\lambda  $ such that
$
	\varphi_\lambda (x_0) = \max_{x \in \Omega_\lambda} |\varphi_\lambda|.
$ Arguing as in (\ref{t0r0}), we have the lower bound

\beq\label{t0r01}
	\frac{\psi_{M \setminus \Omega_\lambda}(t, x_0)}{ \psi_{M \setminus B_{\frac{r_0}{\sqrt{\lambda}}}}(t, x_0)} \geq \alpha.
\eeq

We now observe that by setting $t := \frac{t_0}{\lambda}$ we still have the freedom to choose $ t_0 $. We show that we can select $t_0$ such that (\ref{t0r01}) is violated. To this end set $ t_0 := r_0^2 $ and observe that the upper bound (\ref{eq:Heat-Content-Upper-Bound}) along with (\ref{JKF1}) and Theorem \ref{compare} give:

\begin{align}\label{wtf}
	\frac{\psi_{M \setminus \Omega_\lambda}(\frac{t_0}{\lambda}, x)}{\psi_{M \setminus B_{\frac{r_0}{\sqrt{\lambda}}}}(\frac{t_0}{\lambda}, x)} & \lesssim \frac{1 - e^{-t_0}}{1 - \frac{1}{2^{\nu - 1}\Gamma (\nu + 1)}\sum^\infty_{k = 1}\frac{j_{\nu, k}^{\nu - 1}}{J_{\nu + 1}(j_{\nu, k})}e^{-\frac{j^2_{\nu, k}t_0}{2r_0^2}}} 
	\\ & = \frac{1 - e^{-t_0}}{1 - \frac{1}{2^{\nu - 1}\Gamma (\nu + 1)}\sum^\infty_{k = 1}\frac{j_{\nu, k}^{\nu - 1}}{J_{\nu + 1}(j_{\nu, k})}e^{-\frac{j^2_{\nu, k}}{2}}} \nonumber\\ & = \frac{1 - e^{-t_0}}{\tilde{C}}\nonumber.
\end{align}

Now, we choose $ t_0 = r_0^2 $ small enough, so the last estimate yields a contradiction with (\ref{t0r01}). This proves the theorem. 
\end{proof}

\begin{remark}
	We wish to comment that in the above proof, it is not essential to look at the nodal domain only around the maximum point $x_0$. Given a pre-determined positive constant $\beta$, choose a point $y \in \Omega_\lambda$ such that $\varphi_\lambda (y) \geq \beta \varphi_\lambda (x_0)$. Arguing similarly as in (\ref{stein1}), we see that $\psi_{M \setminus \Omega_\lambda}(t, y) \leq 1 - \beta e^{-t_0}$. Following the computations in (\ref{wtf}), we get a constant $r_0$ (depending on $\beta$) such that $
	\frac{1 - \beta e^{-t_0}}{\tilde{C}} < \alpha$, giving a contradiction.
\end{remark}
\begin{remark}
	Note that sets $ \Sigma $ admitting a Whitney stratification into strata of dimension at most $ n-1 $ (e.g. such as algebraic varieties and semialgebraic sets) are also encompassed in our discussion above.
\end{remark}

\section{Large ball at a max point}\label{LBMP}


In this section we discuss the asymptotic thickness of nodal domains around extremal points of eigenfunctions. More precisely, let us consider a fixed nodal domain $ \Omega_\lambda $ corresponding to the eigenfunction $ \varphi_\lambda $.
Let $ x_0 \in \Omega_\lambda  $ be such that
\begin{equation}
	\varphi_\lambda (x_0) = \max_{x \in \Omega_\lambda} |\varphi_\lambda|.
\end{equation}


In the case $ \dim M = 2$, it was shown in \cite{Man1} that at such maximal points $ x_0 $ one can
fully inscribe a large ball of wavelength radius (i.e $ \sim \frac{1}{\sqrt{\lambda}} $)  into the nodal domain. %It is natural to pose this question for higher dimensions.
In other words for Riemannian surfaces, one has that
\begin{equation}
	\frac{C_1}{\sqrt{\lambda}} \leq \text{inrad }(\Omega_\lambda) \leq \frac{C_2}{\sqrt{\lambda}},
\end{equation}
where $ C_i $ are constants depending only on $ M $. Note that the proof for this case, as carried out in \cite{Man1} by following ideas in ~\cite{NPS}, makes use of essentially $2$-dimensional tools (conformal coordinates and quasi-conformality), which are not available in higher dimensions.
%This question is still open in higher dimensions.

To our knowledge, in higher dimensions the sharpest known bounds on the inner radius of a nodal domain appear in ~\cite{Man2} (Theorem 1.5) and state that:
\begin{equation} \label{eq:Mangoubi-Inrad}
	\frac{C_1}{\lambda^{\alpha(n)}} \leq \text{inrad }(\Omega_\lambda) \leq \frac{C_2}{\sqrt{\lambda}},
\end{equation}
where $ \alpha (n) := \frac{1}{4}(n-1)+\frac{1}{2n} $. A question of current investigation is whether the last lower bound on $ \text{inrad }(\Omega_\lambda) $ in higher dimensions is optimal. 

Here we exploit heat equation and Brownian motion techniques to show that at least, one can expect to ``almost'' inscribe a large ball having radius to the order of $\frac{1}{\sqrt{\lambda}}$, in all dimensions. %We already have the following observations:\newline
%As in (\ref{stein1}), one has
%	\begin{equation}\label{F-K} 
%		\psi_{M \setminus \Omega_\lambda} (t,x_0) \leq 1 - e^{-t\lambda},
%	\end{equation}
%	by a direct application of the Feynman-Kac formula.
	
%	On the other hand, we may choose $ r_0 $, such that
%	\begin{equation}
%		\psi_{M \setminus B_{r_0}} (t', x_0) \geq 1 - \eta,
%		\end{equation}
%where $\eta$ is an arbitrarily small positive number and for convenience we denote $  B_{r_0} := B(x_0, \frac{r_0}{\sqrt{\lambda}}) $. This comes from (\ref{JKF1}). Indeed, the hitting probability of spheres depends only on the ratio between the first hitting time and the corresponding radius of the sphere. So, as commented before, the last inequality is a property of the diffusion process intrinsic to $M$ and is unrelated to any considerations of nodal geometry.
Now we prove Theorem \ref{th:Large-Inscribed-Ball}:%state our theorem:
%\begin{theorem} \label{th:Large-Inscribed-Ball}
%Let $\text{dim }M \geq 3$ and let $ \epsilon_0 > 0 $ be fixed. There exists $ r_0 = r_0 (\epsilon_0) $, such that
%	\begin{equation}
%		\frac{\Vol\left( B_{r_0} \cap \Omega_\lambda \right)}{\Vol\left(B_{r_0}\right)} \geq 1 - \epsilon_0,
%	\end{equation}%
%	where $ B_{r_0} $ denotes $ B\left(x_0, \frac{r_0}{\sqrt{\lambda}}\right) $.
%\end{theorem}

\begin{proof}
	%First, we fix some notation. We denote by $ \psi_K (t,x) $ the probability that a Brownian motion starting at $x$\footnote{CLARIFY!} hits a closed set $ K $ by the time $ t > 0 $.
	%In the previous notation, one may write
	%\begin{equation}
	%	p_t(x) = \psi_{M \setminus \Omega_\lambda} (t,x).
	%\end{equation}
	
	
	We denote $ t' := \frac{t_0}{\lambda} $, and thus $ \psi_{M \setminus \Omega_\lambda} (t',x) \leq 1 - e^{-t_0} $, where $t_0$ is a small constant to be chosen suitably later. %Intuitively, this means the following: it is highly probable that a particle undergoing Brownian motion starting at $x_0$ stays in the nodal domain $ \Omega_\lambda $ for a small amount of time.
	 
	%Observe that $ \psi $ is monotonic with respect to set inclusion. 
	Now, choosing $ t_0 $ small enough, and using monotonicity, we have,
	
	\begin{equation}\label{21}
		\psi_{B_{r_0} \setminus \Omega_\lambda}(t, x_0) < \psi_{M \setminus \Omega_\lambda}(t, x_0) < \epsilon.
	\end{equation}
	
	For convenience, let us denote $E_{r_0} := B_{r_0} \backslash \Omega_\lambda $ - a relatively compact set. Observe that Theorem \ref{compare} applies to open balls and compact subsets contained in open balls. To adapt to the setting of Theorem \ref{compare}, choose a number $r_0^{'} < r_0$ such that $B_{r^{'}_0}$ satisfies $\frac{\text{Vol}(B_{r_0}\setminus B_{r^{'}_0})}{\text{Vol}(B_{r_0})} < \epsilon$. Call $E_{r_0^{'}} := \overline{E_{r_0} \cap B_{r^{'}_0}}$. Observe that proving that $\frac{\text{Vol}(E_{r^{'}_0})}{\text{Vol}(B_{r_0})} < \epsilon$ will imply that $\frac{\text{Vol}(E_{r_0})}{\text{Vol}(B_{r_0})} < 2\epsilon$, which is what we want.
	
	Now, we would like to compare the volumes of the two sets $E_{r^{'}_0}$ and $B_{r_0}$. %Let us choose a point $o \in \pa E_{r_0} \cap \pa B_{r_0}$. Also, lL
	Let $r = \frac{r_0}{\sqrt{\lambda}}$. 
	%Recall that the following inequalities hold for compact sets $ K'\backslash K $:
	%\begin{equation}
%		\capacity(K) \int_0^t \inf_{y \in \partial K} p(s,x,y) ds \leq \psi_{K} (t,x) \leq \capacity(K, K') \int_0^t \sup_{y \in K'\backslash K} p(s,x,y) ds,
%	\end{equation}
%	for $ x \notin K' $. For more details, see
Recall from ~\cite{GS}, Remark 4.1, the following inequality:
\beq\label{Maz}
	c\frac{\text{cap}(E_{r^{'}_0})r^2}{\text{Vol}(B_{r_0})}e^{-C\frac{r^2}{t'}} \leq \psi_{E_{r_0'}}(t', x_0),
\eeq
where $\text{cap}(K)$ denotes the $2$-capacity of the set $K \subset M$, and $0 < t^{'} < 2r^2 $ (see also Equation (3.20) of ~\cite{GS}). Formally, this holds on complete non-compact non-parabolic manifolds, which includes $\RR^n, n \geq 3$. %so we have to adapt its application. If $g_M$ is the metric on $M$, consider $\RR^n$ with the smooth metric $g$, such that $g = g_M$, when $|x| \leq r$ and $g = g_{\RR^n}$ outside a ball of radius $2r$. It is clear that $(\RR^n, g)$ is complete and non-parabolic. 
For bringing in our comparability result Theorem \ref{compare}, we fix the ratio $\frac{t'}{r^2} = \frac{1}{3}$, say, and then choose $t_0$ small enough that (\ref{21}) still works. Now (\ref{Maz}) applies, albeit with a new constant $c$ as determined by the ratio $t/r^2$ and Theorem \ref{compare}. %Formally, this holds for subsets of $ \mathbb{R}^n $ but here we are working in small wavelength balls. This means that one can take a finite atlas $ (U_i, \phi_i) $ with $ \phi_i (U_i) = \mathbb{R}^n $ and choose compact sets $ V_i \subset U_i $ so that $ \cup_i V_i = M $. Moreover, we can arrange that the metric $ g $ is comparable to the Euclidean metric on $ V_i $. It follows that we may consider $ E_{r_0}, B_{r_0} $ as subsets of $ V_i $. By metric comparability the estimates in (\ref{Maz}) still hold with some new appropriate constants, which are independent of $\lambda$.


%Observe that (\ref{Maz}) works only when $0 < t' < 2r^2$. So we can fix the ratio $\frac{t'}{r^2} = \frac{1}{3}$, say, and then choose $t_0$ small enough that (\ref{21}) still works.

Now, to rewrite the capacity term in (\ref{Maz}) in terms of volume, we bring in the following ``isocapacitary inequality'' (see ~\cite{Maz}, Section 2.2.3):
\beq\label{II}
\text{cap}(E_{r^{'}_0}) \geq C'\text{Vol}(E_{r^{'}_0})^{\frac{n - 2}{n}}, n \geq 3,
\eeq
where $C'$ is a constant depending only on the dimension $n$. We note that the isocapacitary inequality (in combination with a suitable Poincare inequality) lies at the heart of the currently optimal inradius estimates, as derived by Mangoubi in \cite{Man2}.

Clearly, (\ref{Maz}) and (\ref{II}) together give
\beq \label{eq:Volume-Ratio}
\left( \frac{\text{Vol}(E_{r^{'}_0})}{\text{Vol}(B_{r_0})} \right)^{\frac{n-2}{n}} \lesssim \frac{\text{cap}(E_{r^{'}_0})r^2}{\text{Vol}(B_{r_0})} \lesssim \psi_{E_{r^{'}_0}}(t, x) < \epsilon.
\eeq
The last inequalities contain constants depending only on $ M $, so by taking $ \epsilon $ even smaller we can arrange $\frac{\text{Vol}(E_{r^{'}_0})}{\text{Vol}(B_{r_0})} < \epsilon_0$ for any initially given $ \epsilon_0 $.
\end{proof}
%\todo{Check Remark}
\begin{remark}
We note that the heat equation method does not distinguish between a general domain and a nodal domain. This means that we cannot rule out the situation where $B(x_0, \frac{r_0}{\sqrt{\lambda}}) \setminus \Omega_\lambda$ is a collection of ``sharp spikes'' entering into $B(x_0, \frac{r_0}{\sqrt{\lambda}})$. Indeed, the probability of a Brownian particle hitting a spike, no matter how ``thin'' it is, or how far from $x_0$ it is, is always non-zero, a fact related to the infinite speed of propagation of heat diffusion. This is consistent with the heuristic discussed in ~\cite{H} and ~\cite{L}.
\end{remark}


 %This is the content of the following
Now we establish Corollary \ref{cor:Large-Ball}. First, we recall the following result, which gives a bound on the asymmetry between the volumes of positivity and negativity sets, as developed in \cite{Man2}:
	\begin{theorem}{\cite{Man2}} \label{th:Asymmetry}
		Let $ B $ be a geodesic ball, so that $ \left( \frac{1}{2}B \cap \{ \varphi_\lambda = 0 \} \right) \neq \emptyset$ with $ \frac{1}{2} B $ denoting the concentric ball of half radius. Then
		\begin{equation}\label{29}
			\frac{\Vol(\{\varphi_\lambda > 0 \}\cap B)}{\Vol(B)} \geq \frac{C}{\lambda^\frac{n-1}{2}}.
		\end{equation}
	\end{theorem} 


\begin{proof}[Proof of Corollary \ref{cor:Large-Ball}]
	It suffices to combine the estimate (\ref{eq:Volume-Ratio}) with (\ref{29}).
	%	More precisely, 
	
	Let $ r:= \frac{r_0}{\sqrt{\lambda}} $ be the radius of the largest inscribed ball in the nodal domain at $ x_0 $. Noting that $ \{ \phi_\lambda < 0  \} \subseteq E_{r_0} $ and combining Theorem \ref{th:Asymmetry} for $ B(x_0, 2r) $ with (\ref{eq:Volume-Ratio}), we get:
	\begin{equation}
		\left( \frac{C}{\lambda^\frac{n-1}{2}} \right)^{\frac{n-2}{n}} \lesssim \left( \frac{\text{Vol}(E_{r_0})}{\text{Vol}(B_{r_0})} \right)^{\frac{n-2}{n}} \leq 1 - e^{-1/3r_0^2}
	\end{equation}
	Expanding the right hand side in Taylor series and rearranging finishes the proof.
\end{proof}

\begin{remark}
	An inspection of the proof of Theorem \ref{th:Large-Inscribed-Ball} reveals that one can take $ \epsilon = r_0^{ \frac{2n}{n-2}} $. In other words,
	the relative volume of the error set $ E_{r_0} $ decays as $ r_0^{ \frac{2n}{n-2}} $ as $ r_0 \rightarrow 0 $. This is slightly better than the scaling
	prescribed by Corollary 2 of \cite{L}.
\end{remark}


%\includegraphics{Spikes}

\section{Appendix 1: Computations using the heat kernel}\label{sec:Heat-Kernel-Comment}

In Section \ref{Perf} we addressed the problem of inscribing a nodal domain $ \Omega_\lambda $ in a tubular neighbourhood around $ \Sigma $.

In this context, an interesting subcase one might also consider is $ \Sigma $ having conical singularities: at its singular points $ \Sigma $ looks locally like $ \mathbb{R}^{n-1-k} \times \partial C^k $ for some $ k=1,\dots, n-1 $, where $ \partial C^k $ denotes the boundary of a generalized cone, i.e. the cone generated by some open set $ D \subseteq \mathbb{S}^{n-1}$.

In this situation a useful tool is an explicit heat kernel formula for generalized cones $ C \subseteq \mathbb{R}^n $. One denotes the associated Dirichlet eigenfunctions and eigenvalues of the generating set $ D $ by $m_j, l_j $ respectively. Using polar coordinates $ x = \rho \theta, y = r \eta $, one has that the heat kernel of $ P_C(t,x,y) $ of the generalized cone $ C $ is given by
\begin{equation} \label{eq:Heat-Kernel-Cones}
	P_C(t,x,y)= \frac{e^{-\frac{\rho^2+r^2}{2t}}}{t(\rho r)^{\frac{n}{2}+1}} \sum_{j=1}^{\infty} I_{\sqrt{l_j+(\frac{n}{2}-1)^2}} (\frac{\rho r}{t}) m_j(\theta) m_j(\eta),
\end{equation}
where $ I_\nu(z) $ denotes the modified Bessel function of order $ \nu $.
For more on the formula (\ref{eq:Heat-Kernel-Cones}) we refer to \cite{BS}. An even more general formula can be found in ~\cite{C}.

The expression for $ P_C(t,x,y) $ provides means for estimating $ p_t(x) $ from below as in Section \ref{Perf}. However, some features of the conical singularity (i.e. the eigenvalues and eigenfunctions $ l_j, m_j $ of the generating set $ D $) enter explicitly in the estimate. Such considerations appear promising in discussing theorems of the following type, for example, and their higher dimensional analogues (see also ~\cite{St}):
\begin{theorem}[Bers, Cheng] Let $n = 2$. If $-\Delta u = \lambda u$, then any nodal set satisfies an interior cone condition with opening angle $\alpha \gtrsim \lambda^{-1/2}$.
\end{theorem}
We leave the detailed discussion to a further note.
%\section{A computation via a Heat Kernel} \label{sec:Heat-Kernel}

%We study a specific type of non-admissible hypersurfaces (in the sense of Section \ref{sec:Intro}), namely hypersurfaces with conical singularities in the following sense:

%\begin{definition} \label{def:Conical-Hypersurface}
%	We say that a hypersurface $\Sigma$ is \textit{conical}, if at its singular points $ \Sigma $ looks locally like $ \mathbb{R}^{n-1-k} \times \partial C^k $ for some $ k=1,\dots, n-1 $, where $ \partial C^k $ denotes the boundary of a generalized cone, i.e. the cone generated by some open set $ D \subseteq \mathbb{S}^{n-1}$.
%\end{definition}

%We show that a sufficiently small wavelength neighbourhood of a conical hypersurface $ \Sigma $ cannot contain a nodal domain $ \Omega_\lambda $, provided that, roughly speaking, the singularities of $ \Sigma $ are not too pointy (in terms of $ \lambda $).

%\begin{proposition}
%	There exists a constant $ c>0 $, such that a nodal domain $ \Omega_\lambda $ cannot be contained in a $N_\epsilon := \frac{c}{\sqrt{\lambda}} $-neighbourhood of $ \Sigma $.
%\end{proposition}

%\begin{proof}

%	Assume to the contrary that a nodal domain $ \Omega_\lambda $ can be inscribed in $ N_\epsilon $, no matter how small $ c $ is.

%\end{proof}
\section{Appendix 2: Stochastic analysis on compact manifolds}\label{benperes}
\subsection{Feynman-Kac formula}\label{F-K}
We begin by stating a Feynman-Kac formula for open connected domains in compact manifolds for the heat equation with Dirichlet boundary conditions. Such formulas seem to be widely known in the community, but since we were unable to find out an explicit reference, we also indicate a line of proof.
\begin{theorem}\label{F-K-T}
Let $M$ be a compact Riemannian manifold. For any open connected $\Omega \subset M$, $f \in L^2(\Omega)$, we have that 
\beq
e^{t\Delta} f(x) = \mathbb{E}_x(f(\omega(t))\phi_\Omega(\omega, t)), t > 0, x \in \Omega,
\eeq
where $\omega (t)$ denotes an element of the probability space of Brownian motions starting at $x$, and we define 
$$
\phi_\Omega(\omega, t) = 
\begin{cases}
1, & \text{if } \omega ([0, t]) \subset \Omega\\
0, & \text{otherwise. }
\end{cases}
$$
\end{theorem}

A proof of Theorem \ref{F-K-T} can be constructed in two steps. First, one proves the corresponding statement when $\Omega = M$. This can be found, for example, in 
~\cite{BP}, Theorem 6.2. One can then combine this with a barrier potential method to prove a corresponding statement for domains $\Omega$ with Lipschitz boundary. Lastly, the extension to domains with no regularity requirements on the boundary is achieved by a standard limiting argument. For details on the last two steps, see ~\cite{T}, Chapter 11, Section 3.

\subsection{Euclidean comparability of hitting probabilities}\label{Eu-comp}
Implicit in many of our calculations is the following heuristic: if the metric is perturbed slightly, hitting probabilities of compact sets by Brownian particles are also perturbed slightly, provided one is looking at small distances $r$ and at small time scales $t = O(r^2)$. Since we are unable to find a reference, here we give a formal statement to suit this heuristic and also indicate a proof. We believe such a statement will also be of independent interest for people interested in stochastic analysis on manifolds.

To describe the set up, let $(M, g)$ be a compact Riemannian manifold and cover $M$ by charts $(U_k, \phi_k)$ such that in these charts $g$ is bi-Lipschitz to the Euclidean metric. Consider an open ball $B(p, r) \subset M$, where $r$ is considered small, and in particular, smaller than the injectivity radius of $M$. Let $B(p, r)$ sit inside a chart $(U, \phi)$ and let $\phi (p) = q$ and $\phi (B(p, r)) = B(q, s) \subset \RR^n$. Let $K$ be a compact set inside $B(p, r)$ and let $K^{'} := \phi(K) \subset B(q, s)$. 

Now, let $\psi_K^M(T, p)$ denote the probability that a Brownian motion on $(M, g)$ started at $p$ and killed at a fixed time $T$ hits $K$ within time $T$. $\psi_{K^{'}}^e(t, q)$ is defined similarly for the standard Brownian motion in $\RR^n$ started at $q$ and killed at the same fixed time $T$. Now, we fix the time $T = cr^2$, where $c$ is a constant. The following is the comparability result:
\begin{theorem}\label{compare}
There exists constants $c_1, c_2$, depending only on $c$ and $M$ such that 
\beq
c_1 \psi_{K^{'}}^e(T, q) \leq \psi_K^M(T, p) \leq c_2 \psi_{K^{'}}^e(T, q).\eeq
\end{theorem}

The proof uses the concept of Martin capacity (see ~\cite{BPP}, Definition 2.1):
\begin{definition}
	Let $\Lambda$ be a set and $\mathcal{B}$ a $\sigma$-field of subsets of $\Lambda$. Given a measurable function $F : \Lambda \times \Lambda \to [0, \infty]$ and a finite measure $\mu$ on $(\Lambda, \Cal{B})$, the $F$-energy of $\mu$ is 
	\[
	I_F(\mu) = \int_{\Lambda}\int_{\Lambda}F(x, y)d\mu(x)d\mu(y).
	\]
	The capacity of $\Lambda$ in the kernel $F$ is 
	\beq
	\text{Cap}_F(\Lambda) = \left[ \inf_\mu I_F(\mu)\right]^{-1},
	\eeq
	where the infimum is over probability measures $\mu$ on $(\Lambda, \Cal{B})$, and by convention, $\infty^{-1} = 0$. 
	\end{definition} 

Now we quote the following general result, which is Theorem 2.2 in ~\cite{BPP}.
\begin{theorem}
	Let $\{X_n\}$ be a transient Markov chain on the countable state space $Y$ with initial state $\rho$ and transition probabilities $p(x, y)$. For any subset $\Lambda$ of $Y$, we have
	\beq
	\frac{1}{2}\text{Cap}_M(\Lambda) \leq \mathbb{P}_\rho[\exists n \geq 0 : X_n \in \Lambda] \leq \text{Cap}_M(\Lambda),\eeq
	where $M$ is the Martin kernel $M(x, y) = \frac{G(x, y)}{G(\rho, y)}$, and $G(x, y)$ denotes the Green's function.
\end{theorem} 

For the special case of Brownian motions, this reduces to (see Proposition 1.1 of ~\cite{BPP} and Theorem 8.24 of ~\cite{MP}):
\begin{theorem}
	Let $\{B(t): 0 \leq t \leq T\}$ be a transient Brownian motion in $\RR^n$ starting from the point $\rho$, and $A \subset D$ be closed, where $D$ is a bounded domain. Then,
	\beq
	\frac{1}{2}\text{Cap}_M(A) \leq \mathbb{P}_\rho\{B(t) \in A \text{ for some } 0 < t \leq T\} \leq \text{Cap}_M(A).\eeq
\end{theorem}

An inspection of the proofs reveals that they go through with basically no changes on a compact Riemannian manifold $M$, when the Brownian motion is killed at a fixed time $T = cr^2$, and the Martin kernel $M(x, y)$ is defined as $\frac{G(x, y)}{G(\rho, y)}$, with $G(x, y)$ being the ``cut-off'' Green's function defined as follows: if $h_M(t, x, y)$ is the heat kernel of $M$,
\[
G(x, y) := \int_0^T h_M(t, x, y)dt.
\]

Now, to state it formally, in our setting, we have
\begin{theorem}
\beq
\frac{1}{2}\text{Cap}_M(K) \leq \psi_K^M(T, p) \leq \text{Cap}_M(K).
\eeq
\end{theorem}

Now, let $h_{\RR^n}(t, x, y)$ denote the heat kernel on $\RR^n$. To prove Theorem \ref{compare}, it suffices to show that for $y \in K$, and $y^{'} = \phi(y) \in K^{'}$, we have constants $C_1, C_2$ (depending on $c$ and $M$) such that 
\beq
C_1\int_0^T h_{\RR^n}(t, q, y^{'})dt \leq \int_0^T h_M(t, p, y)dt \leq C_2 \int_0^T h_{\RR^n}(t, q, y^{'})dt.\eeq

In other words, we need to demonstrate comparability of Green's functions ``cut off'' at time $T = cr^2$. But this follows immediately, using the Gaussian two-sided heat kernel bounds on a compact manifold, and observing that 
\[
\int_0^{cr^2}t^{-\frac{n}{2}}e^{-\frac{r^2}{4t}}\;dt \approx \frac{1}{r^{n-2}}\Gamma\Big(\frac{n}{2}-1,\frac{1}{4c}\Big),\]
where $\Gamma(s,x)$ is the (upper) incomplete Gamma function.


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\end{document}
1602.07110.extracted_bib
[BP]{BP C. B\"{ar and F. Pf\"{affle, {\em Wiener measures on Riemannian manifolds and the Feynman-Kac formula, Mat. Contemp. {40 (2011), 37 - 90.
1602.07110.extracted_bib
[BS]{BS R. Ba\~{nuelos, R. G. Smits, {\em Brownian motion in cones, Probability Theory and Related Fields, {108 (1997), no. 3, Page 299 - 319.
1602.07110.extracted_bib
[BPP]{BPP I. Benjamini, R. Pemantle and Y. Peres, {\em Martin capacity for Markov chains, Ann. Probab., {23 (1995), no. 3, 1332 - 1346.
1602.07110.extracted_bib
[C]{C J. Cheeger, {\em Spectral geometry of singular Riemannian spaces, J. Diff. Geom., {18 (1983), 575 - 657.
1602.07110.extracted_bib
[Ch]{Ch S.-Y. Cheng, {\em Eigenfunctions and nodal sets, Comment. Math. Helv., {51 (1) (1976), 43 - 55.
1602.07110.extracted_bib
[CD]{CD C. Croke and A. Derdzinski, {\em A lower bound for $\lambda_1$ on manifolds with boundary, Comment. Math. Helv. {62 (1987), no. 1, 106–121.
1602.07110.extracted_bib
[GJ]{GJ D. Grieser and D. Jerison, {\em The size of the first eigenvalue of a convex planar domain, J. Amer. Math. Soc., {11 (1998), no. 1, 41 - 72.
1602.07110.extracted_bib
[GS]{GS A. Grigor'yan and L. Saloff-Coste, {\em Hitting probabilities for Brownian motion on Riemannian manifolds, J. Math. Pures Appl., {81 (2002), 115 - 142.
1602.07110.extracted_bib
[H]{H W. Hayman, {\em Some bounds for principal frequency, Applicable Anal., {7 (1977/78), no. 3, 247 - 254.
1602.07110.extracted_bib
[Hs]{Hs E. Hsu, {\em Stochastic analysis on manifolds, Graduate Studies in Mathematics 38, AMS, Providence, RI, 2002.
1602.07110.extracted_bib
[Ke]{Ke J. Kent, {\em Eigenvalue expansions for diffusion hitting times, Z. Wahrsch. Verw. Gebiete, {52 (1980), no. 3, 309 - 319.
1602.07110.extracted_bib
[L]{L E. Lieb, {\em On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math., {74 (1983), no. 3, 441 - 448.
1602.07110.extracted_bib
[Man1]{Man1 D. Mangoubi, {\em On the inner radius of Nodal domains, Canad. Math. Bull. {51 (2008), no. 2, 249 - 260.
1602.07110.extracted_bib
[Man2]{Man2 D. Mangoubi, {\em Local asymmetry and the inner radius of nodal domains, Comm. PDE, {33 (2008), 1611 - 1621.
1602.07110.extracted_bib
[Maz]{Maz V. Maz'ya, {\em Sobolev spaces with applications to elliptic partial differential equations, Grundlehren der Mathematischen Wissenschaften, 342. Springer, Heidelberg, 2011.
1602.07110.extracted_bib
[MS]{MS V. Maz'ya and M. Shubin, {\em Can one see the fundamental frequence of a drum?, Lett. Math. Phys., {74 (2005), 135 - 151.
1602.07110.extracted_bib
[MP]{MP P. M\"{orters and Y. Peres, {\em Brownian motion, With an appendix by Oded Schramm and Wendelin Werner, Cambridge University Press, 2010.
1602.07110.extracted_bib
[NPS]{NPS F. Nazarov, L. Polterovich and M. Sodin, {\em Sign and area in nodal geometry of Laplace eigenfunctions, Amer. J. Math., {127 (2005), no. 4, 879 - 910.
1602.07110.extracted_bib
[St]{St S. Steinerberger, {\em Lower bounds on nodal sets of eigenfunctions via the heat flow, Comm. PDE, {39 (2014), no. 12, 2240 – 2261.
1602.07110.extracted_bib
[T]{T M. Taylor, {\em Partial differential equations. II. Qualitative studies of linear equations, Applied Mathematical Sciences, 116. Springer-Verlag, New York, 1996.
1602.07110.extracted_bib
[Wa]{Wa G. Watson, {\em A treatise on the theory of Bessel functions, 2nd Edition, Cambridge University Press, 1944.