Interior $C^{2}$ estimate for semi-convex solutions to a class of Hessian quotient equations in arbitrary dimensions

TL;DR

This paper establishes interior second derivative estimates for semi-convex solutions to certain Hessian quotient equations in arbitrary dimensions, extending to sum Hessian equations and including rigidity results.

Contribution

It provides new interior $C^{2}$ estimates for Hessian quotient equations and their sum variants under natural conditions, advancing understanding of these nonlinear PDEs.

Findings

Derived interior $C^{2}$ estimates for Hessian quotient equations

Extended results to sum Hessian equations

Established several rigidity theorems

Abstract

In this paper, we study the interior $C^{2}$ estimates for Hessian quotient equations $\frac{\sigma_{3}(D^{2}u)}{\sigma_{l}(D^{2}u)}=1$ for $l=1, 2$, in arbitrary dimensions, under the natural ellipticity and semi-convexity conditions. We further derive analogous results for the corresponding sum Hessian equations. In addition, we establish several rigidity results.