Constant rank theorems for curvature problems via a viscosity approach
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.
Cite
@article{arxiv.2012.03867,
title = {Constant rank theorems for curvature problems via a viscosity approach},
author = {Paul Bryan and Mohammad N. Ivaki and Julian Scheuer},
journal= {arXiv preprint arXiv:2012.03867},
year = {2023}
}
Comments
22 pages