A note on Perelman's no shrinking breather theorem

TL;DR

This paper extends Perelman's no shrinking breather theorem to complete noncompact manifolds using $ ext{L}$-geometry, removing previous technical assumptions and providing a broader understanding of Ricci solitons.

Contribution

It provides a new proof for the noncompact case of Perelman's theorem, utilizing $ ext{L}$-geometry and simplifying prior approaches.

Findings

Proves noncompact shrinking breather is a gradient Ricci soliton

Uses $ ext{L}$-geometry to extend the theorem

Removes technical assumptions from previous proofs

Abstract

As an application of his entropy formula, Perelman proved that every compact shrinking breather is a shrinking gradient Ricci soliton. We give a proof for the complete noncompact case by using Perelman's $\mathcal{L}$-geometry. Our proof follows the argument in Lu and Zheng of constructing an ancient solution, and removes a technical assumption made by them.