Session S33 - Spectral Geometry

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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.}

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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

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