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arXiv · 2012.03867

Constant rank theorems for curvature problems via a viscosity approach

Paul Bryan·Mohammad N. Ivaki·Julian Scheuer
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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.

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Paul Bryan, Mohammad N. Ivaki, Julian Scheuer. 2020-12-07. Constant rank theorems for curvature problems via a viscosity approach. https://doi.org/10.1007/s00526-023-02442-5

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