Harnack inequality for nondivergent parabolic operators on Riemannian manifolds

TL;DR

This paper extends the Krylov-Safonov Harnack inequality to non-divergent parabolic operators on Riemannian manifolds under curvature conditions, providing new proofs and generalizations of classical inequalities.

Contribution

It establishes a Krylov-Safonov Harnack inequality for non-divergent parabolic operators on manifolds, generalizing previous elliptic results and connecting to Li-Yau inequalities.

Findings

Proves Harnack inequality for non-divergent parabolic operators on manifolds.

Provides a new proof of Li-Yau Harnack inequality for heat equations.

Extends curvature condition requirements for Harnack inequalities.

Abstract

We consider second-order linear parabolic operators in non-divergence form that are intrinsically defined on Riemannian manifolds. In the elliptic case, Cabr\'e proved a global Krylov-Safonov Harnack inequality under the assumption that the sectional curvature of the underlying manifold is nonnegative. Later, Kim improved Cabr\'e's result by replacing the curvature condition by a certain condition on the distance function. Assuming essentially the same condition introduced by Kim, we establish Krylov-Safonov Harnack inequality for nonnegative solutions of the non-divergent parabolic equation. This, in particular, gives a new proof for Li-Yau Harnack inequality for positive solutions to the heat equation in a manifold with nonnegative Ricci curvature.