Gradient and Eigenvalue Estimates on the canonical bundle of K\"ahler manifolds

TL;DR

This paper establishes gradient, eigenvalue, and heat kernel estimates for the Hodge Laplacian on canonical bundle sections of K"ahler manifolds, relying solely on Ricci curvature bounds rather than full curvature tensor conditions.

Contribution

It introduces a new Bochner formula for $(m,0)$ forms that depends only on Ricci and scalar curvature, providing sharper estimates under weaker assumptions.

Findings

Derived gradient estimates for the Hodge Laplacian on $(m,0)$ forms.

Established eigenvalue bounds based on Ricci curvature.

Obtained heat kernel estimates with Ricci curvature dependence.

Abstract

We prove certain gradient and eigenvalue estimates, as well as the heat kernel estimates, for the Hodge Laplacian on $(m,0)$ forms, i.e., sections of the canonical bundle of K\"ahler manifolds, where $m$ is the complex dimension of the manifold. Instead of the usual dependence on curvature tensor, our condition depends only on the Ricci curvature bound. The proof is based on a new Bochner type formula for the gradient of $(m, 0)$ forms, which involves only the Ricci curvature and the gradient of the scalar curvature.