Gaussian heat kernel estimates: from functions to forms

TL;DR

This paper establishes conditions under which Gaussian heat kernel estimates for functions extend to one-forms on certain Riemannian manifolds, focusing on Ricci curvature decay without size constraints.

Contribution

It provides new criteria linking Ricci curvature decay to heat kernel estimates for one-forms, expanding understanding of geometric analysis on manifolds.

Findings

Heat kernel estimates transfer from functions to one-forms under specific Ricci decay conditions.

Conditions depend only on decay at infinity, not on Ricci curvature magnitude.

Results apply to complete non-compact Riemannian manifolds with volume doubling property.

Abstract

On a complete non-compact Riemannian manifold satisfying the volume doubling property, we give conditions on the negative part of the Ricci curvature that ensure that, unless there are harmonic one-forms, the Gaussian heat kernel upper estimate on functions transfers to one-forms. These conditions do no entail any constraint on the size of the Ricci curvature, only on its decay at infinity.