Vanishing theorems for $L^2$ harmonic forms on complete Riemannian manifolds

TL;DR

This paper establishes vanishing theorems for $L^2$ harmonic forms on complete Riemannian manifolds, broadening applicability by removing sign and growth restrictions on weight functions, with implications for stable hypersurfaces.

Contribution

It provides new vanishing theorems for $L^2$ harmonic forms without sign or growth restrictions, extending previous results to more general complete Riemannian manifolds.

Findings

Vanishing theorems under weighted Poincaré inequality

Results applicable to complete stable hypersurfaces

Generalization of Li-Wang and Lam's results

Abstract

This paper contains some vanishing theorems for $L^2$ harmonic forms on complete Riemannian manifolds with a weighted Poincar\'e inequality and a certain lower bound of the curvature. The results are in the spirit of Li-Wang and Lam, but without assumptions of sign and growth rate of the weight function, so they can be applied to complete stable hypersurfaces.