Serrin's type Problems in convex cones in Riemannian manifolds

TL;DR

This paper investigates Serrin's overdetermined boundary value problems within convex cones in Riemannian manifolds, establishing rigidity results, geometric characterizations, and extending the analysis to Euclidean cones with drift Laplacian.

Contribution

It introduces new rigidity theorems and geometric characterizations for Serrin's problems in convex cones in Riemannian and Euclidean spaces, including warped product and drift Laplacian cases.

Findings

Rigidity results for overdetermined problems in warped products

Characterization of geodesic ball intersections with cones

Extension to drift Laplacian in Euclidean cones

Abstract

In this work, we discuss several results concerning Serrin's problem in convex cones in Riemannian manifolds. First, we present a rigidity result for an overdetermined problem in a class of warped products with Ricci curvature bounded below. As a consequence, we obtain a rigidity result for Einstein warped products. Next, we derive a soap bubble result and a Heintze-Karcher inequality that characterize the intersection of geodesic balls with cones in these spaces. Finally, we analyze the analogous ovedetermined problem for the drift Laplacian, where the ambient space is a cone in the Euclidean space.