# Surgery and positive Bakry–Émery Ricci curvature

## TL;DR

This paper explores how to maintain positive curvature in certain geometric structures after performing mathematical surgeries.

## Contribution

The paper introduces new local conditions for preserving positive Bakry-Émery Ricci curvature during surgeries.

## Key findings

- Connected sums and surgeries along higher-dimensional spheres preserve positive Bakry-Émery Ricci curvature under local assumptions.
- All closed, simply-connected spin 5-manifolds can have a weighted Riemannian metric with positive Bakry-Émery Ricci curvature.
- These results yield new examples of manifolds with positive Ricci curvature.

## Abstract

We consider the problem of preserving weighted Riemannian metrics of positive Bakry-Émery Ricci curvature along surgery. We establish two theorems of this type: One for connected sums, and one for surgeries along higher-dimensional spheres. In contrast to known surgery results for positive Ricci curvature, these results are local, i.e. we only impose assumptions on the weighted metric locally around the sphere along which the surgery is performed. As application we then show that all closed, simply-connected spin 5-manifolds admit a weighted Riemannian metric of positive Bakry-Émery Ricci curvature. By a result of Lott, this also provides new examples of manifolds with a Riemannian metric of positive Ricci curvature.