A note on the splitting theorem for the weighted measure

TL;DR

This paper investigates the geometric structure of complete manifolds with weighted measures, establishing a splitting theorem under conditions on the weighted Laplacian spectrum and Bakry-Émery Ricci curvature bounds.

Contribution

It extends the splitting theorem to manifolds with finite weighted volume ends under spectral and curvature bounds, analyzing the equality case of prior results.

Findings

Proves a splitting theorem for weighted manifolds with finite volume ends.

Establishes conditions on the spectrum and curvature for manifold splitting.

Analyzes the equality case of a previous theorem by the author.

Abstract

In this paper we study complete manifolds equipped with smooth measures whose spectrum of the weighted Laplacian has an optimal positive lower bound and the $m$-dimensional Bakry-\'Emery Ricci curvature is bounded from below by some negative constant. In particular, we prove a splitting type theorem for complete smooth measure manifolds that have a finite weighted volume end. This result is regarded as a study of the equality case of an author's theorem (J. Math. Anal. Appl. 361 (2010) 10-18).