Geometric rigidity of constant heat flow

TL;DR

This paper characterizes Riemannian manifolds with a constant heat flow boundary condition, showing they are isoparametric tubes around minimal submanifolds, linking geometric properties with overdetermined heat equation problems.

Contribution

It proves that manifolds with the constant flow property are exactly isoparametric tubes around minimal submanifolds, establishing a geometric classification for this overdetermined heat problem.

Findings

Manifolds with constant flow property are isoparametric tubes.

The constant flow property is equivalent to the isoparametric condition under analyticity.

Connections are made between the constant flow property and other overdetermined problems.

Abstract

Let $\Omega$ be a compact Riemannian manifold with smooth boundary and let $u_t$ be the solution of the heat equation on $\Omega$, having constant unit initial data $u_0=1$ and Dirichlet boundary conditions ($u_t=0$ on the boundary, at all times). If at every time $t$ the normal derivative of $u_t$ is a constant function on the boundary, we say that $\Omega$ has the {\it constant flow property}. This gives rise to an overdetermined parabolic problem, and our aim is to classify the manifolds having this property. In fact, if the metric is analytic, we prove that $\Omega$ has the constant flow property if and only if it is an {\it isoparametric tube}, that is, it is a solid tube of constant radius around a closed, smooth, minimal submanifold, with the additional property that all equidistants to the boundary (parallel hypersurfaces) are smooth and have constant mean curvature. Hence, the constant flow property can be viewed as an analytic counterpart to the isoparametric property. Finally, we relate the constant flow property with other overdetermined problems, in particular, the well-known Serrin problem on the mean-exit time function, and discuss a counterexample involving minimal free boundary immersions into Euclidean balls.