Structure of spaces with Bakry-\'Emery Ricci curvature bounded below

TL;DR

This paper investigates the geometric structure of limit spaces of manifolds with Bakry-Émery Ricci curvature bounds, showing tangent cones are metric cones and analyzing singularities, with applications to Kähler-Ricci solitons.

Contribution

It extends Cheeger-Colding techniques to Bakry-Émery Ricci curvature, proving tangent spaces are metric cones and analyzing singularities in the limit space.

Findings

Tangent spaces are metric cones.

Limit space singularities are characterized.

Applications to Kähler-Ricci solitons.

Abstract

In this paper, we explore the limit structure of a sequence of Riemannian manifolds with Bakry-\'Emery Ricci curvature bounded below in the Gromov-Hausdorff topology. By extending the techniques established by Cheeger-Cloding for Riemannian manifolds with Ricci curvature bounded below, we prove that each tangent space at a point of the limit space is a metric cone. We also analyze the singular structure of the limit space analogous to a work of Cheeger-Colding-Tian. Our results will be applied to study the limit space of a sequence of K\"ahler metrics arising from solutions of certain complex Monge-Amp\`ere equations for the existence of K\"ahler-Ricci solitons on a Fano manifold via the continuity method.