Complete Shrinking Ricci Solitons have Finite Fundamental Group

TL;DR

This paper proves that complete shrinking Ricci solitons and related spaces with a positive lower Ricci bound have finite fundamental groups, extending previous compact case results to the complete non-compact setting.

Contribution

It generalizes the finiteness of the fundamental group from compact to complete non-compact shrinking Ricci solitons and related spaces using new analytical techniques.

Findings

Complete shrinking Ricci solitons have finite fundamental group.

Complete smooth metric measure spaces with positive Bakry-Emery tensor also have finite fundamental group.

The proof extends compact case methods to the non-compact setting.

Abstract

We show that if a complete Riemannian manifold supports a vector field such that the Ricci tensor plus the Lie derivative of the metric with respect to the vector field has a positive lower bound, then the fundamental group is finite. In particular, it follows that complete shrinking Ricci solitons and complete smooth metric measure spaces with a positive lower bound on the Bakry-Emery tensor have finite fundamental group. The method of proof is to generalize arguments of Garcia-Rio and Fernandez-Lopez in the compact case.