### Session S33 - Spectral Geometry

## Talks

Monday, July 12, 12:00 ~ 12:20 UTC-3

## The Steklov spectrum, the boundary Laplacian and Hörmander’s rediscovered manuscript

### Iosif Polterovich

#### Université de Montréal, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

How close is the Dirichlet-to-Neumann map to the square root of the corresponding boundary Laplacian? This question has been actively investigated in recent years. Somewhat surprisingly, a lot of techniques involved can be traced back to a newly rediscovered manuscript of Lars Hörmander from the 1950s. We present Hörmander’s approach and its applications, with an emphasis on the asymptotics for Steklov eigenvalues.

Joint work with Alexandre Girouard (Université Laval), Mikhail Karpukhin (Caltech) and Michael Levitin (University of Reading).

Monday, July 12, 12:30 ~ 12:50 UTC-3

## Anisotropic inverse problems for elliptic PDE

### Katya Krupchyk

#### University of California, Irvine , USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk we shall discuss some recent progress on inverse boundary problems for linear and nonlinear elliptic PDE on Riemannian manifolds. In particular, we shall show that the presence of a nonlinearity may help, allowing one to solve inverse problems for nonlinear equations in situations where the linear counterpart is open. We shall also present a solution of the linearized anisotropic Calderon problem on transversally anisotropic manifolds, under the assumption that the transversal manifold is real analytic and satisfies a certain condition related to the geometry of pairs of intersecting geodesics. The argument depends on a construction of Gaussian beam quasimodes on the transversal manifold, with exponentially small errors. This talk is based on joint works with Tony Liimatainen, Mikko Salo, and Gunther Uhlmann.

Monday, July 12, 13:00 ~ 13:20 UTC-3

## Geodesic beams and Weyl remainders

### Jeffrey Galkowski

#### University College London, United Kingdom - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk we discuss quantitative improvements for Weyl remainders under dynamical assumptions on the geodesic flow. We consider a variety of Weyl type remainders including asymptotics for the eigenvalue counting function as well as for the on and off diagonal spectral projector. These improvements are obtained by combining the geodesic beam approach to understanding eigenfunction concentration together with an appropriate decomposition of the spectral projector into quasimodes for the Laplacian. One striking consequence of these estimates is a quantitatively improved Weyl remainder on all product manifolds.

Joint work with Yaiza Canzani (University of North Carolina at Chapel Hill, United States).

Monday, July 12, 14:00 ~ 14:20 UTC-3

## On the spectrum of the Laplace operator of rank one symmetric spaces

### Paolo Piccione

#### Universidade de S\~ao Paulo, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

I will discuss the computation of the first eigenvalue of the Laplace operator of a homogeneous compact rank one symmetric space (CROSS), and give two applications of such computation. The first application is a proof that that isospectral CROSSes are isometric. The second application concerns the stability of the homogeneous metrics as solutions of the Yamabe problem. If time allows, I will also present a description of the full Laplace spectrrum of distance spheres in symmetric spaces of rank one, with applications to a bifurcation problem for CMC hypersurfaces.

Joint work with R. Bettiol (CUNY, Lehman College, USA) and E. Lauret (Univ. Nacional del Sur, Argentina).

Monday, July 12, 14:30 ~ 14:50 UTC-3

## Local bifurcation for Yamabe type equations

### Jimmy Petean

#### CIMAT, Mexico - This email address is being protected from spambots. You need JavaScript enabled to view it.

We consider subcritical Yamabe type equations depending on a real parameter. Local bifurcation from the trivial family of solutions happens when the parameter equals (a fixed multiple of) an eigenvalue of the Laplacian. We will focus in the case of the round sphere and discuss the local structure of the space of solutions at these bifurcation points, which is related to certain integrals involving the corresponding eigenfunctions.

Monday, July 12, 15:00 ~ 15:20 UTC-3

## The Yamabe constants under constraints

### Guillermo Henry

#### Universidad de Buenos Aires-IMAS CONICET, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

Given a compact $n-$dimensional Riemannian manifold $(M,g)$ we say that $U$ is a solution of the Yamabe equation if satisfies (for some constant $c$) the equation $$\frac{4(n-1)}{(n-2)}\Delta_g U+s_gU=c|U|^{\frac{4}{n+2}}U, $$ where $s_g$ is the scalar curvature of $(M,g)$. From a geometric point of view study this equation is interesting because positive solutions are related to constant scalar curvature metrics in the conformal class of $g$. The Yamabe constant is the infimum of the total scalar curvature functional restricted to the conformal class. By the resolution of the celebrated Yamabe Problem this conformal invariant is achieved in any conformal class and it induces a positive solution of the Yamabe equation. On the other hand, the second Yamabe constant, which is the infimum of the second eigenvalues of the conformal Laplacian among the unit volume metrics conformal to $g$ is related to nodal solutions of the Yamabe equation, i.e., sign changing solutions.

In this talk we are going to discuss some results on the existence of solutions of the Yamabe equation obtained by considering the Yamabe constants under certain constraints like isoparametric functions or subgroups of the isometry group.

Tuesday, July 13, 12:00 ~ 12:20 UTC-3

## Do the Hodge spectra detect orbifold singularities?

### Carolyn Gordon

#### Dartmouth College, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

We will address the question: Does the spectrum of the Hodge Laplacian on $p$-forms distinguish closed Riemannian orbifolds with singularities from smooth Riemannian manifolds? This question remains open in the case of the Laplace-Beltrami operator (i.e., the case $p=0$), although many partial results are known. We show that the spectra of the Hodge Laplacians on functions and 1-forms together distinguish manifolds from orbifolds with sufficiently large singular set. We also obtain weaker affirmative results for the spectrum on 1-forms alone and show via counterexamples that some of these results are sharp.

Joint work with Katie Gittins (Durham University), Magda Khalile (Leibniz University), Ingrid Membrillo Solis (University of Southampton), Mary Sandoval (Trinity College) and Elizabeth Stanhope (Lewis & Clark College).

Tuesday, July 13, 12:30 ~ 12:50 UTC-3

## Mutually isospectral Riemannian manifolds

### Benjamin Linowitz

#### Oberlin College, United States of America - This email address is being protected from spambots. You need JavaScript enabled to view it.

An old problem asks, given a natural number $g$, for the maximum number of Riemann surfaces $S_1,...,S_k$ of genus $g$ such that $S_1,...,S_k$ all have the same spectrum of the Laplacian. In this talk we'll report on recent progress on this problem and its natural generalization to locally symmetric spaces.

Joint work with Mikhail Belolipetsky (IMPA, Brazil).

Tuesday, July 13, 13:00 ~ 13:20 UTC-3

## Towards hearing three-dimensional geometric structures

### Craig Sutton

#### Dartmouth College, United States of America - This email address is being protected from spambots. You need JavaScript enabled to view it.

The positive resolution of the geometrization conjecture shows that geometric structures (i.e., complete locally homogeneous metrics) play a special role in our understanding of the taxonomy of three-dimensional manifolds. In light of this, we initiate the exploration of the extent to which three-dimensional geometric structures are determined by their spectra. For example, we find that among locally homogeneous manifolds, closed three-manifolds modeled on six of the eight Thurston geometries are determined up to universal Riemannian cover by their spectra, a result that includes all compact locally symmetric spaces. More generally, we obtain results concerning spaces modeled on ``metrically maximal geometries.''

Joint work with Samuel Lin (Dartmouth College, USA) and Benjamin Schmidt (Michigan State University, USA).

Tuesday, July 13, 13:30 ~ 13:50 UTC-3

## Spectra and representation equivalence for compact symmetric spaces of rank one.

### Roberto J. Miatello

#### FaMAF, Universidad Nacional de Córdoba, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $G$ be a compact Lie group, $K$ a closed subgroup, and let $M=G/K$ endowed with a $G$-invariant Riemannian metric. Given a finite dimensional representation $(\tau,W_\tau)$ of $K$ let $E_\tau$ be the associated hermitian $G$-homogeneous vector bundle over $M$. There is a self-adjoint, second order, elliptic differential operator $\Delta_\tau$ acting on smooth sections of $E_\tau$, defined by the Casimir element $C$ of $G$.

Given a finite subgroup $\Gamma$ of $G$, the quotient $\Gamma\backslash M$ is a compact good orbifold, with a manifold structure in case $\Gamma$ acts freely on $M$. Then $\Gamma \backslash M$ inherits a Riemannian metric on $\Gamma\backslash M$ and $E_\tau$ naturally induces a vector bundle $E_{\tau,\Gamma}$ over $\Gamma\backslash M$, whose sections are identified with the $\Gamma$-invariant sections of $E_\tau$.

The spectrum of the operator $\Delta_{\tau,\Gamma}$, given by the restriction of $\Delta_\tau$ to the space of $\Gamma$-invariant smooth sections of $E_{\tau}$ -- called the $\tau$-spectrum of $\Gamma\backslash M$ -- can be expressed in Lie theoretical terms. Indeed, if we decompose $L^2(\Gamma \backslash G)_\tau =\sum_{\pi \in \widehat G_\tau} n_\Gamma(\pi)V_\pi$, the multiplicity of $\lambda$ in the spectrum of $\Delta_{\tau,\Gamma}$ has a simple linear expression in terms of the multiplicities $n_\Gamma(\pi)$ with $\pi \in \widehat G_\tau$. In particular, the representation of $G$ in $L^2(\Gamma\backslash G)_\tau$ determines the spectrum of the operator $\Delta_{\tau,\Gamma}$. The converse question arises, whether the $\tau$-spectrum $\Gamma\backslash M$ determines the representation $L^2(\Gamma\backslash G)_\tau$ of $G$.

Jointly with Emilio Lauret we have studied the case of compact symmetric spaces of rank one, giving conditions on $G$, $K$ and $\tau$ so that the $\tau$-spectrum determines the representation of $G$ on $L^2(\Gamma\backslash G)_\tau$. In particular, we show the existence of infinitely many $\tau \in \widehat K$ so that the representation-spectral converse holds.

We specially study the case of $p$-form representations, i.e. the irreducible subrepresentations $\tau$ of the representation $\tau_p$ of $K$ on the $p$-exterior power of the complexified cotangent bundle $\bigwedge^p T_C^*M$. We show that for such $\tau$, in most cases $\tau$-isospectrality implies $\tau$-representation equivalence.

Joint work with Emilio A. Lauret (Universidad Nacional del Sur, Bahía Blanca, Argentina).

Tuesday, July 13, 14:30 ~ 14:50 UTC-3

## Polyakov Formulas for conical singularities in two dimensions

### Clara Aldana

#### Universidad del Norte, Colombia - This email address is being protected from spambots. You need JavaScript enabled to view it.

In the talk I will introduce the regularized determinant of the Laplace operator on a Riemannian manifold and will explain the context and the motivation to consider Polyakov's formulas. Then, I will present the formula for surfaces with conical singularities and smooth conformal factors, and for polygonal domains in a Riemannian surface. Time permitting, I will mention the variational Polyakov formula for cones and sectors and how in these cases we can obtain closed formulas for the logarithmic derivative of the determinant of the Laplacian. The results presented in this talk are joint work with Klaus Kirsten and Julie Rowlett, arxiv.org/abs/2010.02776.

Joint work with Klaus Kirsten (Baylor University Waco, US) and Julie Rowlett (Chalmers University of Technology and the University of Gothenburg, Sweden).

Tuesday, July 13, 15:00 ~ 15:20 UTC-3

## Scarring of quasimodes on hyperbolic manifolds

### Lior Silberman

#### The University of British Columbia, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $M$ be a compact hyperbolic manifold. The entropy bounds of Anantharaman et al. restrict the possible invariant measures on $T^1 M$ that can be quantum limits of sequences of eigenfunctions. Weaker versions of the entropy bounds also apply to approximate eigenfuctions ("log-scale quasimodes"), so it is interesting to construct such approximate eigenfunctions which converges to singular measures.

Generalizing work of Brooks (hyperbolic surfaces) and Eswarathasan--Nonnenmacher (hyperbolic geodesics on Riemannian surfaces) we construct sequences of quasimodes on $M$ converging to totally geodesic submanifolds. A diagonal argument then realizes every invariant measure are a limit of quasimodes of fixed logarithmic width.

Joint work with Suresh Eswarathasan (Dalhousie University, Canada).

Tuesday, July 13, 15:30 ~ 15:50 UTC-3

## Eigenvalue bounds for the mixed Steklov problem in two dimensions

### Emily Dryden

#### Bucknell University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

The Steklov problem is an eigenvalue problem for manifolds with boundary; the eigenvalues can also be viewed as the eigenvalues of the Dirichlet-to-Neumann operator, or voltage-to-current map. The question of finding meaningful bounds for these eigenvalues has a long history, beginning with Weinstock's isoperimetric inequality for the lowest nontrivial Steklov eigenvalue of a simply-connected Lipschitz planar domain. We will explore some recent contributions to the story, with an emphasis on results that can be seen in pictures.

Joint work with Teresa Arias-Marco (Universidad de Extremadura, Spain), Carolyn S. Gordon (Dartmouth College, USA), Asma Hassannezhad (University of Bristol, UK), Allie Ray (Birmingham-Southern College, USA) and Elizabeth Stanhope (Lewis & Clark College, USA).

Wednesday, July 21, 16:00 ~ 16:20 UTC-3

## Analytic centrally symmetric plane domains are spectrally determined in that class

### Steve Zelditch

#### Northwestern, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.

The purpose of my talk is to sketch the proof of the statement in the title. To date, there are only 4 classes of plane domains known to be determined by their Dirichlet (or, Neumann) eigenvalues in some specified class of domains: Discs (Kac, 1965) and general ellipses of small eccentricity (Hezari-Z, 2019), among all smooth plane domains; up-down symmetric analytic domains (Z, '09) among all analytic domains satisfying a finite number of conditions; and extremal domains for certain spectral invariants (Marvizi-Melrose, Watanabe). My talk adds a new class: centrally symmetric domains. The proof uses wave invariants in a similar way to that for up-down symmetric domain. Aside from the necessary modifications, the proof contains two new additions. First, it is proved that the finite number of constraints on the analytic domains gives an open-dense set of convex analytic domains when they are assumed convex (and a residual set in general). Second, we exhibit a new duality among billiard maps of domains with a bouncing ball (2-link) orbit. Namely, there are two non-isometric domains whose billiard maps have symplectically equivalent billiard maps around bouncing ball orbits (more precisely, the same Birkhoff normal form). These domains are not isospectral: they can be distinguished by a Maslov index invariant. Joint work with Hamid Hezari.

Joint work with Hamid Hezari, UCI, USA..

Wednesday, July 21, 16:30 ~ 16:50 UTC-3

## Lower bounds for eigenfunction restrictions in lacunary regions

### John Toth

#### McGill, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it..a

Let $(M,g)$ be a compact, real-analytic Riemannian manifold and $u_h \in C^{\omega}(M)$ be a sequence of $L^2$-normalized Laplace eigenfunctions with $(-h^2 \Delta_g - 1) u_h = 0$. We assume that this sequence has a localized defect measure $d\mu$ in the sense that $$ \text{supp} \, \pi_* d\mu = K, \quad M \setminus K \neq \emptyset.$$ Using Carleman estimates in the lacunary region $M \setminus K,$ we show that for any separating hypersurface $H \subset (M\setminus K)$ sufficiently close to $\partial K,$ there exist constants $h_0(H), C_H>0$ such that for $h \in (0, h_0(H)],$ $$ \int_{H} |u_h|^2 d\sigma_H \geq e^{- C_H /h}.$$ Consequently, In the terminology of Toth and Zelditch, all such hypersufaces are good for the eigenfunction sequence $\{ u_h \}.$ This is joint work with Yaiza Canzani.

Joint work with Yaiza Canzani (University of North Carolina, USA).

Wednesday, July 21, 17:00 ~ 17:20 UTC-3

## Internal wave attractors and the spectra of some zeroth-order pseudodifferential operators: a numerical study

### Nilima Nigam

#### Department of Mathematics, Simon Fraser University, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

The propagation of internal gravity waves in stratified media (such as those found in ocean basins and lakes) leads to the development of attractors. These structures accumulate much of the wave energy and can make the fluid flow highly singular. These questions have been the subject of fascinating recent analytical developments by de Verdière & Saint-Raymond, and Zworski and co-workers.

In joint work with Javier Almonacides, we analyze this phenomenon from a numerical analysis perspective. First, we propose a high-accuracy computational method to solve the evolution problem, whose long-term behaviour is known to be non-square-integrable. Then, we use similar tools to discretize the corresponding eigenvalue problem. Since the eigenvalues are embedded in a continuous spectrum, their computation is based on viscous approximations. Finally, we explore the effect that the embedded eigenmodes have in the long-term evolution of the system.

Joint work with Javier Almonacides (Simon Fraser University, Canada).

Wednesday, July 21, 17:30 ~ 17:50 UTC-3

## Filament structure in random plane waves

### Melissa Tacy

#### The University of Auckland, New Zealand - This email address is being protected from spambots. You need JavaScript enabled to view it.

Numerical studies of random plane waves, functions \[u=\sum_{j}c_{j}e^{\frac{i}{h}\langle x,\xi_{j}\rangle}\] where the coefficients $c_{j}$ are chosen ``at random'', have detected an apparent filament structure. The waves appear enhanced along straight lines. There has been significant difference of opinion as to whether this structure is indeed a failure to equidistribute, numerical artefact or an illusion created by the human desire to see patterns. In this talk I will present some recent results that go some way to answering the question. We study the behaviour of a random variable $G(x,\xi)=||P_{(x,\xi)}u||_{L^{2}}$ where $P_{(x,\xi)}$ is a semiclassical localiser at Planck scale around $(x,\xi)$ and show that $G(x,\xi)$ fails to equidistribute. This suggests that the observed filament structure is a configuration space reflection of the phase space concentrations.

## Posters

## A Friedland-Hayman inequality for convex cones

### Thomas Beck

#### Fordham University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

The Friedland-Hayman inequality concerns the growth rates of homogeneous, harmonic functions with Dirichlet boundary conditions on complementary cones dividing Euclidean space into two parts. In this talk, we will describe a variant of this inequality where one divides a convex cone into two parts, with Neumann conditions on the boundary of the cone, and Dirichlet conditions on the shared interface. This inequality plays a crucial role in the boundary regularity of a two-phase free boundary problem in a convex domain. The proof, which uses a variant of Caffarelli's contraction theorem for the Brenier optimal transport mapping, allows us to characterize the case of equality.

Joint work with David Jerison (MIT, USA) and Sarah Raynor (Wake Forest University, USA).

## An introduction to Toeplitz quantization

### Alix Deleporte

#### Universite Paris Saclay, France - This email address is being protected from spambots. You need JavaScript enabled to view it.

Toeplitz quantization associates, to a function on a phase space, a family of self-adjoint operators, indexed by a semiclassical parameter. The resulting Berezin-Toeplitz operators encompass usual pseudodifferential operators as well as quantum spin systems and the quantization of Arnold's catmap on the torus.

The geometric ingredients of Toeplitz quantization are a symplectic manifold (phase space) with a "compatible" Riemannian metric (or, equivalently, a compatible complex structure), and an associated magnetic Laplacian. The lowest eigenvalue of this magnetic Laplacian is highly degenerate, and the associated spectral projector, named the Szegö or Bergman projector, allows one to define Toeplitz quantization.

In this introductory talk, I will introduce the Toeplitz picture and its modern challenges; this will be motivated by applications to spectral theory, geometry, PDEs, and theoretical physics.

## Fourier restriction on hyperbolic manifolds

### Xiaolong Han

#### California State University, Northridge, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

The Fourier restriction phenomenon asks whether one can meaningfully restrict the Fourier transform of a function onto a hypersurface (such as a sphere) in the frequency space. Stein’s restriction conjectures state that the Fourier transform of an Lp function restricts to a well-defined Lq function on the sphere, for appropriate ranges of exponents p and q. While the full conjecture remains open, Tomas and Stein in the 1970s proved the case when q=2. Via the spectral measure, the Tomas-Stein restriction estimates have been proved in geometries other than the Euclidean spaces such as asymptotically conic or hyperbolic manifolds, all of which require that there is no geodesic trapping, i.e., all geodesics extend to infinity. In this talk, we study how the restriction estimates are influenced by this trapping condition. We present the first examples of manifolds with geodesic trapping for which the Tomas-Stein restriction estimates hold.

## Uniform Sobolev estimates on compact manifolds involving singular potentials.

### Xiaoqi Huang

#### Johns Hopkins University, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.

We will discuss generalizations of the uniform Sobolev inequalities of Kenig, Ruiz and Sogge for Euclidean spaces and Dos Santos Ferreira, Kenig and Salo for compact Riemannian manifolds involving critically singular potentials V ∈ Ln/2. We shall also discuss the analogous improved quasimode estimates as well as analogues of the improved uniform Sobolev estimates involving such potentials under certain geometric conditions.

Joint work with Matthew D. Blair (University of New Mexico), Yannick Sire (Johns Hopkins University) and Christopher Sogge (Johns Hopkins University).

## The Two-Point Weyl Law on Manifolds without Conjugate Points

### Blake Keeler

#### McGill University, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk, we discuss the asymptotic behavior of the spectral function of the Laplace-Beltrami operator on a compact Riemannian manifold $M$ with no conjugate points. The spectral function, denoted $\Pi_\lambda(x,y),$ is defined as the Schwartz kernel of the orthogonal projection from $L^2(M)$ onto the eigenspaces with eigenvalue at most $\lambda^2$. In the regime where $(x,y)$ is restricted to a sufficiently small compact neighborhood of the diagonal in $M\times M$, we obtain a uniform logarithmic improvement in the remainder of the asymptotic expansion for $\Pi_\lambda$ and its derivatives of all orders. This generalizes a result of B\'erard that established an on-diagonal estimate for $\Pi_\lambda(x,x)$ without derivatives. Furthermore, when $(x,y)$ avoids a compact neighborhood of the diagonal, we obtain the same logarithmic improvement in the standard upper bound for the derivatives of $\Pi_\lambda$. We also discuss an application of these results to the study of monochromatic random waves.

## Sounds of symmetry

### Julie Rowlett

#### Chalmers University and the University of Gothenburg, Sweden - This email address is being protected from spambots. You need JavaScript enabled to view it.

If you are attending this special session, then you probably know the famous question `Can one hear the shape of a drum?' as well as its answer. Due to the physical interpretation, we say that quantities determined by the Laplace spectrum can be heard. Such quantities are called spectral invariants. Here I'll present a small collection of results, from both my research as well as others, that show that in some contexts, symmetry can be heard.

Joint work with Results presented include joint works with C. Aldana and Z. Lu.

## The heat content functional and isoparametric foliations

### Alessandro Savo

#### Sapienza Universita' di Roma, Italy - This email address is being protected from spambots. You need JavaScript enabled to view it.

On a smooth bounded domain $\Omega$ in a Riemannian manifold one considers the heat content function $H_{\Omega}(t)$, which is the total heat energy at time $t>0$ of the domain $\Omega$, assuming that the initial temperature is constant, equal to $1$ everywhere on $\Omega$, and that the boundary is subject to absolute refrigeration at all times (Dirichlet boundary conditions).

One expects that geometry heavily affects heat diffusion, and that domains with many symmetries will enjoy special properties, i.e. will be extremal for the functional which associates to a domain its heat content function at a fixed time $t$. For example it is known by symmetrization methods (Burchard and Shmuckenschlager 2001) that, among all domains with fixed volume in a constant curvature space form $M$, geodesic balls realize the absolute maximum of $H_{\Omega}(t)$, for all fixed $t$: when the volume is fixed, geodesic balls will minimize the dispersion of heat due to boundary refrigration.

In this talk, we focus on understanding the geometry of {\it critical domains} for the heat content functional, by first computing its first variation at all times $t$ on an arbitrary Riemannian manifold $M$. It (easily) turns out that on Euclidean space, hyperbolic space, or the {\it hemisphere}, the only critical domains are geodesic balls (which are in fact absolute maximums by the above results).

A bit surprisingly, however, one can show that, already on the whole sphere, there are plenty of domains which are critical but are not isometric to geodesic balls: these are saddle points for the functional, and are domains bounded by {\it isoparametric hypersurfaces}, that is, hypersurfaces which have constant principal curvatures but are not umbilic. Any isoparametric hypersurface gives rise to an {\it isoparametric foliation} of the domain it bounds: that is, a foliation by smooth hypersurfaces all parallel to $\partial\Omega$, all having constant mean curvature, and collapsing to exactly one (minimal) leaf of higher codimension, the focal submanifold of the foliation.

The study of isoparametric hypersurfaces was initiated by Cartan in the 30's of last century, and is a fascinating field which uses algebraic and topological tools; the classification problem took many efforts and has been completed only recently.

Our main result, which we want to discuss in the talk, states that in fact domains bounded by an isoparametric hypersurface are {\it the only} spherical domains which are critical for the heat content functional at any fixed time $t$ (at least when the boundary is connected).

This results comes out of a more general rigidity result, which shows that criticality at all times actually forces the existence of an isoparametric foliation on the domain. In what follows, we say that $\Omega$ is an {\it isoparametric tube around a submanifold $P$} if $\Omega$ is a smooth, solid tube around $P$, such that all hypersurfaces at constant distance to $P$ are smooth and have constant mean curvature. It is seen that on the round sphere this is equivalent to the condition that the boundary is an isoparametric hypersurface.

{\bf Theorem} {\it Let $\Omega$ be a smooth bounded domain in an analytic Riemannian manifold. Then $\Omega$ is critical for the heat content functional $H_{\Omega}(t)$, at every fixed time $t$, for deformations keeping the inner volume constant, if and only if $\Omega$ is an isoparametric tube around a minimal submanifold.}

\smallskip

The same classification holds for the functional given by the $k$-th torsional rigidity, based on a hierarchy of mean-exit time functions which are of interest in probability and potential theory.

The conclusion is that isoparametric foliations play an interesting role in variational geometry and overdetermined PDE's. This role is not evident in Euclidean space, because the only (compact) isoparametric foliation of $\mathbb R^n$ is given by the family of concentric spheres, by the celebrated Alexandrov theorem.

Alessandro Savo, {On the heat content functional and its critical domains} arXiv: 2010.05860 (2021)

Alessandro Savo, {Geometric rigidity of constant heat flow} Calc. Var. 57 (6) 2018, p1-26

## On static manifolds and related critical spaces with zero radial Weyl curvature

### Emanuel Mendonça Viana

#### Instituto Federal de Educação, Ciência e Tecnologia do Ceará - IFCE, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

The aim of this paper is to study compact Riemannian manifolds $(M,\,g)$ that admit a non-constant solution to the system of equations $$-\Delta f\, g+Hess f-fRic=\mu Ric+\lambda g,$$ where $Ric$ is the Ricci tensor of $g$ whereas $\mu$ and $\lambda$ are two real parameters. More precisely, under the assumption that $(M,\,g)$ has zero radial Weyl curvature, this means that the interior product of $\nabla f$ with the Weyl tensor $W$ is zero, we shall provide the complete classification for the following structures: positive static triples, critical metrics of volume functional and critical metrics of the total scalar curvature functional.

The article can be found at https://doi.org/10.1007/s00605-019-01365-8

Joint work with Abdênago Alves de Barros (Universidade Federal do Ceará - UFC, Departamento de Matemática), Halyson Irene Baltazar (Universidade Federal do Piauí - UFPI, Departamento de Matemática) and Rondinelle Marcolino Batista (Universidade Federal do Piauí - UFPI, Departamento de Matemática).

## Pointwise Weyl Laws for Schrodinger operators with singular potentials

### Cheng Zhang

#### University of Rochester, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

We consider the Schr\"odinger operators $H_V=-\Delta_g+V$ with singular potentials $V$ on general $n$-dimensional Riemannian manifolds and study whether various forms of pointwise Weyl law remain valid under this perturbation. First, we prove that the pointwise Weyl law holds for potentials in the Kato class, which is the minimal assumption to ensure that $H_V$ is essentially self-adjoint and bounded from below or has favorable heat kernel bounds. This generalizes the 3-dimensional results by Frank-Sabin to any dimensions. Second, we show that the pointwise Weyl law with the standard sharp error term $O(\lambda^{n-1})$ holds for potentials in $L^n(M)$. This extends the classical results of Avakumovi\'c, Levitan and H\"ormander by obtaining the same error term in the pointwise Weyl laws for the Schr\"odinger operators with rough potentials. Both of the results are expected be sharp by the examples constructed in Frank-Sabin.

Joint work with Xiaoqi Huang, Johns Hopkins University.