Gradient estimate for eigenforms of Hodge Laplacian

TL;DR

This paper derives a gradient estimate for eigenforms of the Hodge Laplacian on closed manifolds, providing a tool for analyzing their properties and applications to heat kernel estimates.

Contribution

It introduces a new gradient estimate for eigenforms of the Hodge Laplacian based on geometric bounds, with direct applications to heat kernel estimation.

Findings

Gradient estimate depends on dimension, volume, diameter, curvature

Sharp heat kernel estimate derived from the gradient bound

Applicable to linear combinations of eigenforms

Abstract

In this paper, we derive a gradient estimate for the linear combinations of eigenforms of the Hodge Laplacian on a closed manifold. The estimate is given in terms of the dimension, volume, diameter and curvature bound of the manifold. As an application, we obtain directly a sharp estimate for the heat kernel of the Hodge Laplacian.