Bounded and $L^2$ Harmonic Forms on Universal Covers

TL;DR

This paper explores the relationship between curvature conditions and the existence of bounded and $L^2$ harmonic forms on universal covers, providing topological insights and vanishing theorems for compact manifolds.

Contribution

It establishes new links between curvature positivity, harmonic forms, and topology, including conditions that lead to vanishing theorems and examples with mixed curvature.

Findings

Positivity of the Weitzenbock curvature term relates to harmonic form existence.

Certain curvature pinching conditions imply vanishing theorems.

Constructed examples of manifolds with negative sectional curvature satisfying vanishing conditions.

Abstract

We relate the positivity of the curvature term in the Weitzenbock formula for the Laplacian on p-forms on a complete manifold to the existence of bounded and $L^2$ harmonic forms. In the case where the manifold is the universal cover of a compact manifold, we obtain topological and geometric information about the compact manifold. For example, we show that a compact manifold cannot admit one metric with pinched negative curvature and another metric with positive Weitzenbock term on two-forms. Many of these results can be thought of as differential form analogues of Myers' theorem. We also give pinching conditions on certain sums of sectional curvatures which imply the positivity of the curvature term, and hence yield vanishing theorems. In particular, we construct a compact manifold with planes of negative sectional curvature at each point and which satisfies the hypothesis of our vanishing theorems.