Harmonic Forms on Manifolds with Non-Negative Bakry-\'Emery-Ricci Curvature

TL;DR

This paper proves that on certain smooth metric measure spaces with non-negative Bakry-Émery-Ricci curvature, the existence of non-trivial weighted harmonic one-forms implies finite weighted volume and a splitting of the universal cover.

Contribution

It establishes a splitting theorem and volume finiteness result for manifolds with non-negative Bakry-Émery-Ricci curvature based on harmonic form properties.

Findings

Weighted volume of the manifold is finite under the given conditions.

Universal cover splits isometrically as a product with the real line.

Non-trivial weighted harmonic one-forms imply geometric and topological constraints.

Abstract

In this paper we prove that on a complete smooth metric measure space with non-negative Bakry-\'Emery-Ricci curvature if the space of weighted L^2 harmonic one-forms is non-trivial then the weighted volume of the manifold is finite and universal cover of the manifold splits isometrically as the product of the real line with an hypersurface.