Constant rank theorems for curvature problems via a viscosity approach

TL;DR

This paper introduces a viscosity-based method to prove constant rank theorems for curvature problems, offering simpler proofs and extending results to new geometric contexts.

Contribution

It generalizes a differential inequality in a viscosity sense to the subtrace, enabling elementary proofs and new applications in geometric curvature equations.

Findings

Provided new elementary proofs for constant rank theorems

Extended constant rank results to non-homogeneous curvature equations

Introduced a viscosity approach to handle eigenvalue regularity issues

Abstract

An important set of theorems in geometric analysis consists of constant rank theorems for a wide variety of curvature problems. In this paper, for geometric curvature problems in compact and non-compact settings, we provide new proofs which are both elementary and short. Moreover, we employ our method to obtain constant rank theorems for homogeneous and non-homogeneous curvature equations in new geometric settings. One of the essential ingredients for our method is a generalization of a differential inequality in a viscosity sense satisfied by the smallest eigenvalue of a linear map (Brendle-Choi-Daskalopoulos, Acta Math. 219(2017): 1-16) to the one for the subtrace. The viscosity approach provides a concise way to work around the well known technical hurdle that eigenvalues are only Lipschitz in general. This paves the way for a simple induction argument.