Relationship between sectional curvature and null spaces of Lichnerowicz-type Laplacians and their smallest eigenvalues

TL;DR

This paper investigates the relationship between sectional curvature and the null spaces of various Laplacians on Riemannian manifolds, establishing positivity conditions and eigenvalue estimates under curvature pinching assumptions.

Contribution

It proves positivity of the curvature operator of the second kind under sectional curvature and Ricci curvature conditions, and provides new vanishing theorems and eigenvalue estimates for Laplacians.

Findings

Curvature operator of the second kind is positive under certain curvature conditions.

Vanishing theorems for null spaces of Laplacians are established.

Lower bounds for the smallest eigenvalues of Laplacians are derived.

Abstract

The first variant of this article contained a fatal error. Therefore, we publish second version our paper. In the present paper, we prove that the curvature operator of the second kind of a Riemannian manifold is strictly positive if its sectional curvature is strictly positive and the Ricci curvature suitably pinched. In addition, we prove several vanishing theorems for null spaces of the Lichnerowicz, Sampson, and Hodge-de Rham Laplacians and find estimates for their lowest eigenvalues on compact (without boundary) Riemannian manifolds with sectional pinched curvature.