Curvature estimates for the level set of spatial quasiconcave solutions to a class of parabolic equations

TL;DR

This paper establishes a constant rank theorem for the second fundamental form of convex level surfaces of solutions to certain parabolic equations, providing geometric curvature bounds under structural conditions.

Contribution

It introduces a novel constant rank theorem for the second fundamental form of level surfaces in solutions to a class of parabolic equations, with geometric curvature estimates.

Findings

Proved a constant rank theorem for the second fundamental form.

Derived a geometric lower bound for principal curvature.

Applied structural conditions to ensure curvature estimates.

Abstract

We prove a constant rank theorem for the second fundamental form of the spatial convex level surfaces of solutions to equations $u_t=F(\n^2u, \n u, u, t)$ under a structural condition, and give a geometric lower bound of the principal curvature of the spatial level surfaces.