Stationary isothermic surfaces in Euclidean 3-space

TL;DR

This paper classifies stationary isothermic surfaces in three-dimensional space, showing they must be either parallel planes or co-axial cylinders if they do not intersect the boundary of the domain, using methods from differential geometry and analysis.

Contribution

It completes the classification of such surfaces under specified conditions, introducing a new connection with transnormal functions and analyzing uniformly dense domains.

Findings

Stationary isothermic surfaces are either parallel planes or co-axial cylinders.

The classification applies when the surface does not intersect the boundary of the domain.

The proof involves techniques from constant mean curvature surface theory and asymptotic analysis.

Abstract

Let $\Omega$ be a domain in $\mathbb R^3$ with $\partial\Omega = \partial\left(\mathbb R^3\setminus \overline{\Omega}\right)$, where $\partial\Omega$ is unbounded and connected, and let $u$ be the solution of the Cauchy problem for the heat equation $\partial_t u= \Delta u$ over $\mathbb R^3,$ where the initial data is the characteristic function of the set $\Omega^c = \mathbb R^3\setminus \Omega$. We show that, if there exists a stationary isothermic surface $\Gamma$ of $u$ with $\Gamma \cap \partial\Omega = \varnothing$, then both $\partial\Omega$ and $\Gamma$ must be either parallel planes or co-axial circular cylinders . This theorem completes the classification of stationary isothermic surfaces in the case that $\Gamma\cap\partial\Omega=\varnothing$ and $\partial\Omega$ is unbounded. To prove this result, we establish a similar theorem for {\it uniformly dense domains } in $\mathbb R^3$, a notion that was introduced by Magnanini, Prajapat \& Sakaguchi in \cite{MPS2006tams}. In the proof, we use methods from the theory of surfaces with constant mean curvature, combined with a careful analysis of certain asymptotic expansions and a surprising connection with the theory of transnormal functions.