A rigidity theorem for Kolmogorov-type operators

TL;DR

This paper generalizes a classical geometric rigidity theorem, originally for Newtonian potentials, to a broader class of Kolmogorov-type partial differential equations, revealing that certain potential proportionality conditions imply spherical symmetry.

Contribution

It extends the Suzuki--Watson theorem from caloric equations to a wider class of Kolmogorov-type PDEs, establishing a new rigidity result.

Findings

Theorem applies to a broader class of PDEs beyond classical Newtonian and caloric cases.

Proves that potential proportionality outside a domain implies the domain is a sphere.

Unifies and generalizes previous rigidity results for different PDE settings.

Abstract

Let $D\subseteq \mathbb{R}^n$, $n\geq 3$, be a bounded open set and let $x_0\in D$. Assume that the Newtonian potential of $D$ is proportional outside $D$ to the Newtonian potential of a mass concentrated at $\{x_0\}.$ Then $D$ is a Euclidean ball centered at $x_0$. This Theorem, proved by Aharonov, Shiffer and Zalcman in 1981, was extended to the caloric setting by Suzuki and Watson in 2001. In this note, we show that Suzuki--Watson Theorem is a particular case of a more general rigidity result related to a class of Kolmogorov-type PDEs.