Non-preservation of concavity properties by the Dirichlet heat flow on Riemannian manifolds

TL;DR

This paper demonstrates that the Dirichlet heat flow does not preserve concavity properties in Riemannian manifolds unless the curvature is zero, highlighting the influence of curvature on heat flow behavior.

Contribution

It establishes a curvature-dependent criterion for the preservation of concavity properties under the Dirichlet heat flow on Riemannian manifolds.

Findings

Concavity is not preserved unless sectional curvature vanishes.

The result links curvature to heat flow properties.

Provides conditions for concavity preservation in geometric analysis.

Abstract

We prove that no concavity properties are preserved by the Dirichlet heat flow in a totally convex domain of a Riemannian manifold unless the sectional curvature vanishes everywhere on the domain.