# Integral Curvature Bounds and Bounded Diameter with Bakry--Emery Ricci Tensor

## TL;DR

This paper extends classical geometric theorems to weighted Riemannian manifolds with bounded Bakry--Emery Ricci tensor and potential function, establishing compactness results under these conditions.

## Contribution

It generalizes Myers' theorem to manifolds with a weighted measure and bounded Bakry--Emery Ricci tensor, providing new compactness criteria.

## Key findings

- Proves a generalized Myers compactness theorem for weighted manifolds.
- Establishes diameter bounds under Bakry--Emery Ricci curvature conditions.
- Shows boundedness of the potential function f is crucial for the results.

## Abstract

For Riemannian manifolds with a smooth measure $(M, g, e^{-f}dv_{g})$, we prove a generalized Myers compactness theorem when Bakry--Emery Ricci tensor is bounded from below and $f$ is bounded.