Symmetry problems on stationary isothermic surfaces in Euclidean spaces

TL;DR

This paper investigates the symmetry properties of stationary isothermic surfaces in Euclidean spaces, showing that such surfaces must possess certain symmetries if they are stationary under heat flow conditions.

Contribution

It establishes a link between stationary isothermic surfaces and their inherent symmetries in Euclidean spaces, extending understanding of geometric properties related to heat flow.

Findings

Stationary isothermic surfaces exhibit specific symmetries.

The symmetry results apply to smooth hypersurfaces in Euclidean spaces.

The work connects heat flow behavior with geometric symmetry properties.

Abstract

Let $S$ be a smooth hypersurface properly embedded in $\mathbb R^N$ with $N \geq 3$ and consider its tubular neighborhood $\mathcal N$. We show that, if a heat flow over $\mathcal N$ with appropriate initial and boundary conditions has $S$ as a stationary isothermic surface, then $S$ must have some sort of symmetry.