On space-time quasiconcave solutions of the heat equation

TL;DR

This paper establishes a constant rank theorem for space-time level sets of quasiconcave solutions to the heat equation, leading to new convexity results for these level sets in convex rings.

Contribution

It introduces a novel constant rank theorem for the second fundamental form of space-time level sets of quasiconcave solutions to the heat equation, advancing convexity analysis.

Findings

Proved a constant rank theorem for the second fundamental form.

Derived strict convexity results for level sets in convex rings.

Reviewed related constant rank techniques for harmonic functions and convex solutions.

Abstract

In this paper we first obtain a constant rank theorem for the second fundamental form of the space-time level sets of a space-time quasiconcave solution of the heat equation. Utilizing this constant rank theorem, we can obtain some strictly convexity results of the spatial and space-time level sets of the space-time quasiconcave solution of the heat equation in a convex ring. To explain our ideas and for completeness, we also review the constant rank theorem technique for the space-time Hessian of space-time convex solution of heat equation and for the second fundamental form of the convex level sets for harmonic function.