Laplacian cut-offs, porous and fast diffusion on manifolds and other applications

TL;DR

This paper develops new tools for analysis on manifolds with Ricci curvature bounds, enabling advances in diffusion processes, gradient estimates, and operator properties without topological restrictions.

Contribution

It introduces a method to construct controlled cut-off functions on manifolds with Ricci bounds, generalizes the Li-Yau gradient estimate, and applies these to diffusion and operator analysis.

Findings

Constructed exhaustion functions with controlled gradient and Laplacian.

Generalized Li-Yau gradient estimate for manifolds with Ricci bounds.

Applied cut-offs to study diffusion equations and operator self-adjointness.

Abstract

We construct exhaustion and cut-off functions with controlled gradient and Laplacian on manifolds with Ricci curvature bounded from below by a (possibly unbounded) nonpositive function of the distance from a fixed reference point, without any assumptions on the topology or the injectivity radius. Along the way we prove a generalization of the Li-Yau gradient estimate which is of independent interest. We then apply our cut-offs to the study of the fast and porous media diffusion, of $L^q$-properties of the gradient and of the self-adjointness of Schroedinger-type operators.