Surgery and positive Bakry-\'Emery Ricci curvature

TL;DR

This paper proves that positive Bakry-Émery Ricci curvature can be preserved under certain surgical modifications of manifolds, leading to new examples of manifolds with positive Ricci curvature.

Contribution

It establishes local surgery theorems for preserving positive Bakry-Émery Ricci curvature, extending known results for positive Ricci curvature.

Findings

Connected sums preserve positive Bakry-Émery Ricci curvature.

Surgeries along higher-dimensional spheres preserve positive Bakry-Émery Ricci curvature.

All closed, simply-connected spin 5-manifolds admit metrics with positive Bakry-Émery Ricci curvature.

Abstract

We consider the problem of preserving weighted Riemannian metrics of positive Bakry-\'Emery 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-\'Emery Ricci curvature. By a result of Lott, this also provides new examples of manifolds with a Riemannian metric of positive Ricci curvature.