Heat flow within convex sets

TL;DR

This paper proves that heat flow solutions within convex sets on Riemannian manifolds remain inside the set until boundary conditions are met, highlighting the preservation of convexity under heat diffusion.

Contribution

It establishes a new convexity preservation property for heat flow solutions on Riemannian manifolds with boundary.

Findings

Heat flow solutions stay within convex sets until boundary images are reached.

Convexity preservation holds for solutions with boundary conditions.

Results extend understanding of heat flow behavior in geometric contexts.

Abstract

A solution to the heat equation between Riemannian manifolds, where the domain is compact and possibly has boundary, will not leave a compact and locally convex set before the image of the boundary does.