Rotational symmetry of complete shrinking gradient Yamabe solitons

TL;DR

This paper proves that certain complete shrinking gradient Yamabe solitons are rotationally symmetric under specific curvature conditions, confirming a Yamabe-soliton version of Perelman's conjecture.

Contribution

It establishes the rotational symmetry of nontrivial complete shrinking gradient Yamabe solitons under optimal curvature bounds, resolving an analogue of Perelman's conjecture.

Findings

Proves rotational symmetry under scalar curvature bounds

Shows the assumption is optimal in higher dimensions

Resolves the Yamabe-soliton analogue of Perelman's conjecture

Abstract

In this paper, we show that any nontrivial complete shrinking gradient Yamabe soliton whose scalar curvature is bounded below by the soliton constant everywhere and is strictly greater than the constant at some point is rotationally symmetric. This assumption is optimal for higher dimensions. This result resolves the Yamabe-soliton analogue of Perelman's conjecture.