$L^2_f$ harmonic 1-forms on smooth metric measure spaces with positive $\lambda_1(\Delta_f)$

TL;DR

This paper investigates vanishing and splitting phenomena of harmonic 1-forms on smooth metric measure spaces under various curvature bounds related to the first eigenvalue of the weighted Laplacian, extending previous results.

Contribution

It extends existing vanishing and splitting theorems to broader curvature conditions involving the first eigenvalue of the weighted Laplacian on smooth metric measure spaces.

Findings

Vanishing results for harmonic 1-forms under certain curvature bounds.

Splitting theorems related to the spectrum of the weighted Laplacian.

Generalizations of classical results to weighted manifolds with negative Bakry-Émery-Ricci curvature.

Abstract

In this paper, we study vanishing and splitting results on a complete smooth metric measure space $(M^n,g,\mathrm{e}^{-f}\mathrm{d}v)$ with various negative $m$-Bakry-\'Emery-Ricci curvature lower bounds in terms of the first spectrum $\lambda_1(\Delta_f)$ of the weighted Laplacian $\Delta_f$, i.e. $\mathrm{Ric}_{m,n}\geq -a\lambda_1(\Delta_f)-b$ for $0<a\leq\dfrac{m}{m-1}, b\geq0$. In particular, we consider three main cases for different $a$ and $b$ with or without conditions on $\lambda_1(\Delta_f)$. These results are extensions of Dung and Vieira, and weighted generalizations of Li-Wang, Dung-Sung and Vieira.