Topology of gradient Ricci shrinkers via weighted $L^2$ cohomology

TL;DR

This paper investigates the topology of smooth gradient Ricci shrinkers using weighted $L^2$ cohomology, providing bounds, vanishing results, and a Hodge theorem, with extensions to mean curvature flow shrinkers.

Contribution

It introduces new topological bounds and theorems for Ricci shrinkers through weighted $L^2$ cohomology, extending classical results to a broader class of geometric flows.

Findings

Established upper bounds for Betti numbers

Proved a vanishing theorem for cohomology

Derived a dichotomy for the number of ends

Abstract

This paper proves several topological results for smooth gradient Ricci shrinkers. We establish upper bounds for the Betti numbers, a vanishing theorem for cohomology, and a dichotomy for the number of ends. We also prove a full Hodge theorem for a large class of shrinkers. The methods are based on weighted $L^2$ cohomology and extend to self-shrinkers of the mean curvature flow.