# Rigidity of closed metric measure spaces with nonnegative curvature

## TL;DR

This paper proves that among closed smooth metric measure spaces with nonnegative Bakry-Émery Ricci curvature, only the circle has the optimal spectral upper bound for the weighted Laplacian, extending previous manifold results.

## Contribution

It establishes a rigidity result characterizing the circle as the unique space with optimal spectral bound under nonnegative curvature conditions.

## Key findings

- Only the circle achieves the optimal spectral upper bound.
- Extension of Hang-Wang's manifold result to metric measure spaces.
- Provides new insights into the structure of spaces with nonnegative Bakry-Émery Ricci curvature.

## Abstract

We show that one-dimensional circle is the only case for closed smooth metric measure spaces with nonnegative Bakry-\'{E}mery Ricci curvature whose spectrum of the weighted Laplacian has an optimal positive upper bound. This result extends the work of Hang-Wang in the manifold case (Int. Math. Res. Not. 18 (2007), Art. ID rnm064, 9pp).