# New proofs of Perelman's theorem on shrinking Breathers in Ricci flow

## TL;DR

This paper presents two novel proofs demonstrating that shrinking breathers in Ricci flow on closed manifolds are necessarily gradient Ricci solitons, utilizing properties of singularity models and ancient solutions.

## Contribution

It introduces two new proofs of Perelman's theorem, expanding the understanding of shrinking breathers in Ricci flow through alternative approaches.

## Key findings

- Shrinking breathers are gradient Ricci solitons.
- Singularity models of type I are shrinking gradient Ricci solitons.
- Rescaled limits of non-collapsed type I ancient solutions are shrinking gradient Ricci solitons.

## Abstract

We give two new proofs of Perelman's theorem that shrinking breathers of Ricci flow on closed manifolds are gradient Ricci solitons, using the fact that the singularity models of type I solutions are shrinking gradient Ricci solitons and the fact that non-collapsed type I ancient solutions have rescaled limits being shrinking gradient Ricci solitons.