A note on the strong maximum principle for fully nonlinear equations on Riemannian manifolds

TL;DR

This paper studies strong maximum and minimum principles for fully nonlinear second order equations on Riemannian manifolds, extending classical results to a broad class of operators and geometric settings.

Contribution

It establishes new strong maximum principles for nonlinear equations on Riemannian manifolds, including operators like Pucci's and those related to p-Laplacian and mean curvature.

Findings

Validates strong maximum principles for a wide class of operators

Derives new comparison principles on manifolds with nonnegative sectional curvature

Extends classical PDE results to geometric settings

Abstract

We investigate strong maximum (and minimum) principles for fully nonlinear second order equations on Riemannian manifolds that are non-totally degenerate and satisfy appropriate scaling conditions. Our results apply to a large class of nonlinear operators, among which Pucci's extremal operators, some singular operators like those modeled on the $p$- and $\infty$-Laplacian, and mean curvature type problems. As a byproduct, we establish new strong comparison principles for some second order uniformly elliptic problems when the manifold has nonnegative sectional curvature.