Rigidity of an overdetermined heat equation and minimal helicoids in space-forms

TL;DR

This paper characterizes domains with constant boundary temperature in overdetermined heat equations across various space-forms, revealing their boundaries are minimal surfaces like planes, helicoids, or special tori, depending on the geometry.

Contribution

It proves that such domains must have minimal boundary surfaces and classifies these surfaces in space-forms, extending previous Euclidean results to spherical and hyperbolic geometries.

Findings

Domains with constant boundary temperature have minimal boundary surfaces.

In $ ext{S}^3$, such domains are bounded by totally geodesic surfaces or Clifford tori.

In $ ext{H}^3$, they are bounded by totally geodesic surfaces or minimal hyperbolic helicoids.

Abstract

Let $M$ be a Riemannian manifold and $\Omega$ a smooth domain of $M$. We study the following heat diffusion problem: assume that the initial temperature is equal to $1$, uniformly on $\Omega$, and is $0$ on its complement. Heat will then flow away from $\Omega$ to its complement, and we are interested in the temperature on the boundary of $\Omega$ at all positive times $t>0$. In particular we ask: are there domains for which the temperature at the boundary is a constant $c$, for all positive times $t$ and for all points of the boundary? If they exist, what can we say about their geometry? This is a typical example of overdetermined heat equation. It is readily seen that if $c$ exists it must be $\frac 12$, and domains with constant boundary temperature will be said to have the $\frac 12$-property. Previous work by \cite{MPS06} and \cite{CSU23} show that, on $\mathbb R^3$, the only such domains (up to congruences) have boundary which is a plane or (a bit surprisingly) the right helicoid. In this paper we first show that, in great generality, the boundary of a $\frac 12$-domain must be minimal; we then extend (with a different proof) the above classification from $\mathbb R^3$ to the other $3$-dimensional space-forms. We prove that, in $\mathbb S^3$, $\frac 12$-domains are bounded by a totally geodesic surface or the Clifford torus, and in the hyperbolic space $\mathbb H^3$ are bounded by a totally geodesic surface or by an (embedded) minimal hyperbolic helicoid. %(there is a one-parameter family of such surfaces) As a by-product, we extend (with a different proof) a result by Nitsche on uniformly dense domains from $\mathbb R^3$ to $3$-dimensional space-forms.