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\title{The isospectral problem for $p$-widths: an application of {Zoll} metrics}
\alttitle{Le problÃ¨me isospectral pour les $p$-largeurs : une application des mÃ©triques de {Zoll}}

\author{\firstname{Jared} \lastname{Marx-Kuo}}
\address{450 Jane Stanford Way, Stanford, United States of America}
\email{jmarxkuo@gmail.com}
\thanks{The author was supported by a Stanford SGF fellowship and an NSF grant number DGE-1656518}
\CDRGrant[NSF]{DGE-1656518}

\subjclass{53C22}

\keywords{\kwd{Min-max} \kwd{$p$-widths} \kwd{Zoll} \kwd{geodesics}}
\altkeywords{\kwd{Min-max} \kwd{$p$-largeurs} \kwd{Zoll} \kwd{gÃ©odesiques}}

\begin{abstract}
We pose the isospectral problem for the $p$-widths: is a Riemannian manifold $(M^{n+1}, g)$ uniquely determined by its $p$-widths, $\bigl\{\omega_p(M,g)\bigr\}_{p=1}^{\infty}$? We construct many counterexamples on $S^2$ using Zoll metrics and the fact that geodesic $p$-widths are given by unions of immersed geodesics.
\end{abstract}

\begin{altabstract}
Nous posons le problÃ¨me isospectral pour les $p$-largeurs : une variÃ©tÃ© Riemannienne $(M^{n+1}, g)$ est-elle uniquement dÃ©terminÃ©e par ses $p$-largeurs, $\smash{\bigl\{\omega_p(M,g)\bigr\}_{p=1}^{\infty}}\,$? Nous construisons de nombreux contre-exemples sur $S^2$ en utilisant les mÃ©triques de Zoll et le fait que les $p$-largeurs gÃ©odÃ©siques sont donnÃ©es par des unions de gÃ©odÃ©siques immergÃ©es.
\end{altabstract}

\begin{document}

\maketitle

\section{Introduction}

In~\cite{gromov2006dimension}, Gromov introduced the \emph{$p$-widths}, $\{\omega_p\}$, of a Riemannian manifold, $(M^{n+1}, g)$, as an analogue of the spectrum of the Laplacian, $\{\lambda_k\}$, the discrete collection of eigenvalues. Intuitively, one replaces the Rayleigh quotient in the definition of eigenvalues with the area functional in the definition of $\omega_p$ -- see Section~\ref{background}, as well as~\cite{gromov2010singularities,gromov2002isoperimetry, guth2009minimax,fraser2020applications} for more background. The $p$-widths have proven to be extremely useful due to the Almgren--Pitts/Marques--Neves Morse theory program for the area functional (see~\cite{almgren1965theory,pitts2014existence,marques2014min,marques2023morse,marques2015morse, marques2021morse,almgren1962homotopy,zhou2020multiplicity}). In~\cite{liokumovich2018weyl}, Liokumovich--Marques--Neves demonstrated that $\{\omega_p\}$ satisfy a Weyl law, which later led to a resolution of Yau's conjecture~\cite{yau1982problem}: any closed three-dimensional manifold must contain an infinite number of immersed minimal surfaces. In some sense, the resolution was stronger than expected:
\begin{theorem}[\cite{chodosh2020minimal,song2018existence,marques2019equidistribution, marques2017existence,irie2018density}]
On any closed manifold $(M^{n+1}, g)$ with $3 \leq n + 1 \leq 7$, there exist infinitely many embedded minimal hypersurfaces.
\end{theorem}

We emphasize the result of \emph{embedded} surfaces over \emph{immersed} ones, and also note that embedded minimal hypersurfaces are automatically smooth when $3 \leq n+1 \leq 7$. We recall the following timeline: Marques--Neves~\cite{marques2017existence} first proved the result in the $\Ric_g &gt; 0$ setting, Irie--Marques--Neves~\cite{irie2018density} proved the result for generic metrics, Chodosh--Mantoulidis~\cite{chodosh2020minimal} proved the result using the Allen--Cahn equation for a generic metric when $n + 1 = 3$, and Song~\cite{song2018existence} proved the result for non-generic metrics. We also note the work of Li~\cite{li2023existence} who showed the existence of infinitely many immersed minimal surfaces with optimal regularity when $n + 1 \geq 8$. We refer the reader to the introduction of~\cite{chodosh2023p} for a more thorough overview of the history and applications of the $p$-widths.

Recall the isospectral problem for the Laplacian:
\begin{enonce}{Problem}\label{LaplacianIsospectral}
Is a closed manifold, $(M^{n+1}, g)$, uniquely determined (up to isometry) by its discrete spectrum, $\{\lambda_k\}$?
\end{enonce}
Informally, this is known as ``can you hear the shape of a drum?'' and was made famous by Kac~\cite{kac1966can}. In~\cite{milnor1964eigenvalues}, Milnor constructed two $16$-dimensional tori which are isospectral but not isometric, providing the first resolution of Problem~\ref{LaplacianIsospectral}. Gordon--Webb--Wolpert~\cite{gordon1992isospectral} constructed simpler examples within the class of polygons in $\R^2$. By contrast, Berger~\cite{berger1966spectre} showed that $S^2$ is spectrally rigid (see also Hersch~\cite{hersch1970quatre} who deduced the same from only the first $3$ eigenvalues).
 
Given the analogy between $\{\omega_p\}$ and $\{\lambda_p\}$, it is natural to pose the \emph{$p$-width isospectral problem}:
\begin{enonce}{Problem}\label{pWidthIsospectral}
Is a closed manifold, $(M^{n+1}, g)$, uniquely determined (up to isometry) by the values $\omega_p(g)$?
\end{enonce}
We show that the answer is ``no'':
\begin{theorem}\label{constantWidthThm}
For any $p &gt; 0$, $\omega_p$ is constant on any fixed connected component of the space of Zoll metrics on $S^2$.
\end{theorem}
As a corollary, we can compute the $p$-widths explicitly for new metrics.

\begin{corollary}\label{IsoSpectralThm}
Let $g$ be any Zoll metric on $S^2$ which lies in the connected component of $g_{\round}$ in the space of Zoll metrics on $S^2$. Then $\omega_p(g) = \omega_p(g_{\round}) = 2 \pi \lfloor \sqrt{p} \rfloor$ for all $p$.
\end{corollary}
The above gives rise to a continuous deformation of $p$-width isospectral metrics (compare~\cite{gord1984iso}). In particular, any odd function $f \colon S^2 \to \R$ gives rise to a $1$-parameter family of geometrically distinct Zoll metrics near the round metric (see~\cite{guillemin1976radon}). The $p$-widths of $(S^2, g_{\round})$ were computed explicitly in~\cite{chodosh2023p}, and this was the first example of the $p$-width spectrum being known for all $p$. Corollary~\ref{IsoSpectralThm} now gives uncountably more manifolds where the full $p$-width spectrum is known.

The proof of Theorem~\ref{constantWidthThm} uses the defining properties of Zoll metrics on $S^2$, which are metrics for which all geodesics (with multiplicity one) are simple and of length $2 \pi$ (note this choice of length, as opposed to any constant $c$, agrees with~\cite{guillemin1976radon}). These were first constructed by Zoll~\cite{zoll1901ueber}, and we refer the reader to~\cite{guillemin1976radon,funk1911flachen,gromoll1981metrics,besse2012manifolds} for a by no means complete list of background sources. The proof utilizes a recent result of Chodosh--Mantoulidis~\cite{chodosh2023p}, which states that each $\omega_p(M^2, g)$ is achieved by a union of closed, potentially immersed geodesics. 

This paper is organized as follows: in Section~\ref{background}, we briefly define the $p$-widths, and recall the results of Chodosh--Mantoulidis~\cite{chodosh2023p}. We also define Zoll metrics and list some relevant properties. In Section~\ref{MainThmSec} we prove the main theorem, and in Section~\ref{OpenSec} we pose some open questions.

\section{Background} \label{background}

We recall the essential notation from the min-max theory of Almgren--Pitts~\cite{almgren1962homotopy} and Marques--Neves~\cite{MN2020CETSurvey}. Let $M$ be an $(n+1)$-dimensional manifold, let $X \subseteq [0,1]^k$ denote a cubical subcomplex of the $k$-dimensional unit cube (cf.~\cite[Section~2.2]{MN2020CETSurvey}), and let $\mathcal{Z}_n(M; \Z_2)$ denote the set of $n$-cycles mod $\Z_2$ -- informally, a cycle is the boundary of a nice (Caccioppoli) open set. In~\cite{almgren1962homotopy}, Almgren showed that $\mathcal{Z}_n(M; \Z_2)$ is weakly homotopic to $\mathbb{R} \mathbb{P}^{\infty}$. We let $\overline{\lambda} \in H^1\parens[\big]{\mathcal{Z}_n(M; \Z_2), \Z_2} \cong \Z_2$ denote the generator of this group, so that $\overline{\lambda}^p$ generates $H^p\parens[\big]{\mathcal{Z}_n(M; \Z_2), \Z_2} \cong \Z_2$. For $T \in \mathcal{Z}_n(M; \Z_2)$, there is a natural associated varifold to $T$ and we let $\norm{T}$ denote the corresponding Radon measure, with $\norm{T}(U)$ denoting the integral of the measure over $U$. We further let $\mathbf{M}(T) = \norm{T}(M)$.
\begin{definition}[{\cite[Definition~4.1]{marques2017existence}}]\label{defi:sweepout}
A map $\Phi \colon X\to \mathcal{Z}_{n}(M;\Z_{2})$ is a \emph{$p$-sweepout} if it is continuous (with the standard flat norm topology on $\mathcal{Z}_n(M; \Z_2)$) and $\Phi^{*}(\overline\lambda^{p}) \neq 0 \in H^p(X)$.
\end{definition}

\begin{definition}[{\cite[Section~3.3]{marques2017existence}}]\label{defi:no.concentration.of.mass}
A map $\Phi \colon X \to \mathcal{Z}_n(M; \Z_2)$ is said to have \emph{no concentration of mass} if
\[
\lim_{r \to 0} \sup \braces[\bigg]{\norm{\Phi(x)}\parens[\big]{B_r(p)} \;:\; x \in X, \; p \in M} = 0.
\]
\end{definition}

\begin{definition}[\cite{gromov2002isoperimetry,guth2009minimax}] \label{defi:p-width}
We define $\mathcal{P}_{p} = \mathcal{P}_{p}(M)$ to be the set of all $p$-sweepouts, out of any cubical subcomplex $X$, with no concentration of mass. The \emph{$p$-width} of $(M,g)$ is
\[
\omega_{p}(M,g) \coloneqq \inf_{\Phi \in \mathcal{P}_{p}} \sup\braces[\bigg]{\mathbf{M}\parens[\big]{\Phi(x)} \;:\; x \in \domain(\Phi)}.
\]
\end{definition}
To see the analogy between $\omega_p$ and $\lambda_p$, consider the following characterization of $\lambda_p(M)$ as
\[
\lambda_p(M) \coloneqq \inf_{\substack{V \subseteq C^{\infty}(M) \\ \dim V = p+1}} \sup_{f \in V \backslash \{0\}} \frac{\int_M \abs{\n f}^2}{\int_M f^2}.
\]
Note the parallels between the min-max definitions of $\omega_p, \lambda_p$, with $\Phi^{*}(\overline\lambda^{p}) \neq 0$ being analogous to $\dim(V) = p+1$.

Having defined the $p$-widths, one expects from the min-max theory of Almgren--Pitts~\cite{almgren1965theory, pitts2014existence} that $\omega_p = \sum_{i} \mathbf{M}(V_i)$ where $V_i$ are stationary varifolds for $\mathbf{M}$. When $3 \leq n+1 \leq 7$, the regularity theory of Schoen--Simon~\cite{schoen1981regularity} shows that such varifolds are induced by smooth minimal hypersurfaces, potentially with multiplicity (see also~\cite[Section~3]{marques2015morse}). However, when $n + 1 = 2$, the $V_i$ may correspond to geodesic networks, as opposed to unions of closed geodesics (cf.~\cite[Remark~1.1]{marques2015morse} and~\cite[Section~5]{chodosh2023p}, as well as~\cite[Figure~1]{marx2024index} for a visual). In~\cite{chodosh2023p}, Chodosh--Mantoulidis used the Allen--Cahn equation to prove that the $p$-widths can always be achieved by a union of closed geodesics:
\begin{theorem}[{\cite[Theorem~1.2]{chodosh2023p}}]\label{ChoManPWidths}
Let $(M^2, g)$ be a closed Riemannian manifold. For every $p \in \Z^+$, there exists a collection of primitive, closed geodesics $\{\sigma_{p,j}\}$ such that
\[
\omega_p(M, g) = \sum_{j = 1}^{N(p)} m_j \cdot L_g(\sigma_{p,j}), \quad m_j \in \Z^+.
\]
\end{theorem}
Here, ``primitive'' means that the geodesic is connected and transversed with multiplicity one. They also showed:
\begin{theorem}[{\cite[Theorem~1.4]{chodosh2023p}}]\label{SphereWidths}
For $g_0$ the unit round metric on $S^2$
\[
\forall p \in \N, \quad \omega_p(S^2, g_0) = 2 \pi \lfloor \sqrt{p} \rfloor.
\]
\end{theorem}
We now collect relevant facts about Zoll metrics on $S^2$.

\begin{definition}\label{ZollMetric}
A metric, $g$, on $S^2$is called a \emph{Zoll metric} if all of the primitive geodesics are closed and of length $2 \pi$.
\end{definition}
Zoll~\cite{zoll1901ueber}, Funk~\cite{funk1911flachen}, Guillemin~\cite{guillemin1976radon}, and Gromoll--Grove~\cite{gromoll1981metrics} each contributed significantly to the theory of Zoll metrics. In particular, Gromoll--Grove showed that all of the primitive geodesics on a Zoll surface are simple. Furthermore, the combined work of Guillemin and Funk demonstrated that the tangent space of Zoll metrics at the round metric is given by odd functions $f \colon S^2 \to \R$. We also mention a higher dimensional analogue of these metrics constructed by Ambrozio--Marques--Neves~\cite{ambrozio2021riemannian}.
 
Let $\mathcal{Z}(S^2)$ denote the space of Zoll metrics on $S^2$, $\mathcal{Z}_{\alpha}$, an arbitrary connected component of $\mathcal{Z}(S^2)$, and $\mathcal{Z}_0$, the connected component containing $g_0$, the unit round metric.

\section{Proof of main theorem} \label{MainThmSec}

Let $\mathcal{M}_{\infty}(S^2)$ denote the set of all smooth metrics on $S^2$, and let $g_0$ denote the round metric. For fixed $p$, $\omega_p \colon \mathcal{M}_{\infty}(S^2) \to \R$ is continuous due to the following lemma of Marques--Neves--Song:
\begin{lemma}[{\cite[Lemma~1]{marques2019equidistribution}}]\label{WidthContinuity}
Let $C_1 &lt; C_2$. There exists a $K(C_1, C_2)$, such that
\[
\abs{\omega_p(g) - \omega_p(g')} \leq K(C_1, C_2) p^{1/2} \norm{g - g'}_{C^0(g_0)}
\]
for any $g, g'$ such that $C_1 g_0 \leq g \leq C_2 g_0$ and $C_1 g_0 \leq g' \leq C_2 g_0$.
\end{lemma}
Take any Zoll metric, $g$, on $S^2$. Applying Theorem~\ref{ChoManPWidths}, we have
\[
\begin{split}
\omega_p(S^2, g)	&amp; = \sum_{j = 1}^{N(p)} m_j \cdot L_{g}(\sigma_{p,j}) \\
				&amp; = 2 \pi \sum_{j = 1}^{N(p)} m_j \in 2 \pi \Z^{+}
\end{split}
\]
since each $\sigma_{p,j}$ is primitive and $L_{g}(\sigma_{p,j}) = 2 \pi$ by definition of Zoll metric. Let $\mathcal{Z}_{\alpha}$ denote a fixed, connected component of $\mathcal{Z}$. Consider the restriction of $\omega_p \colon \mathcal{Z}_{\alpha} \to \R$. Its image is discrete, lying inside $2\pi \N$, but $\omega_p$ is continuous in each component due to Lemma~\ref{WidthContinuity}, thus the map is constant on connected components, proving Theorem~\ref{constantWidthThm}.

\section{Open questions} \label{OpenSec}

Having shown a counterexample to $p$-width spectral rigidity, it would be interesting to know any of the following:
\begin{itemize}
\item Do there exist counterexamples to the isospectral problem in dimensions~$3 \leq n + 1 \leq 7$? The minimal hypersurfaces which achieve the $p$-widths behave differently (e.g.~are embedded and disjoint) in these dimensions compared to $n+1 = 2$, where self-intersecting geodesics, or two geodesics intersecting may occur. See~\cite{marx2024index, aiex2016width} which highlight the differences between multiplicity, immersion vs.~embeddedness, and index bounds of the underlying minimal objects in dimensions $n+1 = 2$ vs.~$3 \leq n + 1 \leq 7$.
\item Are there other surfaces for which $(\Sigma^2, g)$ is $p$-width spectrally rigid up to isometry? We note that shortly after the release of this work, Ambrozio--Marques--Neves~\cite{ambrozio2024rigidity} demonstrated that $\mathbb{R} \mathbb{P}^2$ with the round metric is $p$-width spectrally rigid.
\item Do there exist other reasonable geometric conditions that one can impose on a manifold so that isospectral rigidity holds?
\end{itemize}

\section*{Acknowledgments}

The author thanks Otis Chodosh for his conversations on $p$-widths and Lucas Ambrozio for his conversations on Zoll metrics. The author is also thankful to the referee for their in-depth and thoughtful feedback.

\section*{Declaration of interests}

The author does not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and has declared no affiliations other than their research organizations.

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