A uniqueness theorem for asymptotically cylindrical shrinking Ricci solitons

TL;DR

This paper establishes a uniqueness result for asymptotically cylindrical shrinking Ricci solitons, showing that such solitons matching a standard cylinder at infinity are essentially the same as the cylinder or its quotient.

Contribution

It proves a rigidity theorem for shrinking Ricci solitons that are asymptotic to cylinders, extending understanding of their global geometric structure.

Findings

Shrinking gradient Ricci solitons matching cylindrical metrics at infinity are isometric to the cylinder.

Complete solitons are globally isometric to the cylinder or its a7-quotient.

The result applies to solitons on manifolds with ends asymptotic to cylinders.

Abstract

We prove that a shrinking gradient Ricci soliton which agrees to infinite order at spatial infinity with one of the standard cylindrical metrics on $S^k\times \RR^{n-k}$ for $k\geq 2$ along some end must be isometric to the cylinder on that end. When the underlying manifold is complete, it must be globally isometric either to the cylinder or (when $k=n-1$) to its $\ZZ_2$-quotient.