Constant rank theorems for curvature problems via a viscosity approach

Paul Bryan; Mohammad N. Ivaki; Julian Scheuer

doi:10.1007/s00526-023-02442-5

Constant rank theorems for curvature problems via a viscosity approach

Paul Bryan, Mohammad N. Ivaki, Julian Scheuer^*

^*Corresponding author for this work

School of Mathematical and Physical Sciences

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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 et al. (Acta Math 219:1–16, 2017) 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.

Original language	English
Article number	98
Pages (from-to)	1-19
Number of pages	19
Journal	Calculus of Variations and Partial Differential Equations
Volume	62
Issue number	3
DOIs	https://doi.org/10.1007/s00526-023-02442-5
Publication status	Published - Apr 2023

Bibliographical note

Copyright © 2023, The Author(s). Version archived for private and non-commercial use with the permission of the author/s and according to publisher conditions. For further rights please contact the publisher.

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Constant rank theorems for curvature problems via a viscosity approach

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Analysis of fully non-linear geometric problems and differential equations