On the regular-convexity of Ricci shrinker limit spaces

TL;DR

This paper investigates the geometric structure of Ricci shrinker limit spaces, establishing their regular parts are convex and extending the regular-singular decomposition framework to these limits.

Contribution

It introduces a regular-singular decomposition for Ricci shrinker limits and proves the convexity of their regular parts, extending Cheeger-Colding and Colding-Naber's ideas.

Findings

Regular part of Ricci shrinker limits is convex

Established a regular-singular decomposition for these limits

Extended geometric analysis techniques to Ricci shrinker limits

Abstract

In this paper, we study the structure of the pointed-Gromov-Hausdorff limits of sequences of Ricci shrinkers. We define a regular-singular decomposition following the work of Cheeger-Colding for manifolds with a uniform Ricci curvature lower bound, and prove that the regular part of any Ricci shrinker limit space is convex, inspired by Colding-Naber's original idea of parabolic smoothing of the distance functions.