Integral pinched shrinking Ricci solitons

TL;DR

This paper proves that certain compact gradient shrinking Ricci solitons in dimensions 4 to 6, satisfying an integral pinching condition, are isometric to quotients of round spheres, using algebraic curvature estimates and inequalities.

Contribution

It establishes a rigidity result for integral pinched shrinking Ricci solitons in low dimensions, extending the classification of such geometric structures.

Findings

Proves rigidity of integral pinched shrinking Ricci solitons in dimensions 4-6.

Shows these solitons are isometric to quotients of round spheres.

Uses sharp algebraic curvature estimates and Yamabe-Sobolev inequality.

Abstract

We prove that a $n$-dimensional, $4 \leq n \leq 6$, compact gradient shrinking Ricci soliton satisfying a $L^{n/2}$-pinching condition is isometric to a quotient of the round $\mathbb{S}^{n}$. The proof relies mainly on sharp algebraic curvature estimates, the Yamabe-Sobolev inequality and an improved rigidity result for integral pinched Einstein metrics.