The existence of $(\mathbf{p}, k)$-convex hypersurfaces for a class of Hessian quotient type curvature equations

TL;DR

This paper proves the existence and uniqueness of certain convex hypersurfaces satisfying Hessian quotient curvature equations, using a priori estimates, the continuity method, and inverse convexity properties.

Contribution

It introduces new existence and uniqueness results for $( extbf{p}, k)$-convex hypersurfaces related to Hessian quotient curvature equations, leveraging inverse convexity.

Findings

Existence and uniqueness of $( extbf{p}, k)$-convex hypersurfaces established.

Proved a constant rank theorem for the operator $rac{\sigma_k}{\sigma_l}( abla^2 u)$.

Derived solutions for both nonhomogeneous and homogeneous curvature equations.

Abstract

This article investigates the existence of closed, star-shaped hypersurfaces for a class of Hessian quotient type curvature equations, in which the operator $\frac{\sigma_k}{\sigma_l}(\Lambda)$ arising in these equations can be viewed as a generalization of the classical Hessian quotient operator. By combining a priori estimates with the continuity method, we establish the existence and uniqueness of $(\mathbf{p}, k)$-convex hypersurfaces for both nonhomogeneous and homogeneous equations of this type. Furthermore, by exploiting the recently discovered ``inverse convexity'' property of the operator $\frac{\sigma_k}{\sigma_l}(\Lambda)$, we prove a constant rank theorem and thereby obtain the existence and uniqueness of strictly convex solutions to these curvature equations.