On the global structure of conformal gradient solitons with nonnegative Ricci tensor

TL;DR

This paper classifies the global structure of complete conformal gradient solitons with nonnegative Ricci tensor, showing they are either product spaces, Euclidean space, or spheres, and establishes rotational symmetry for certain Yamabe-type solitons.

Contribution

It provides a complete classification of conformal gradient solitons with nonnegative Ricci tensor and proves rotational symmetry for specific Yamabe-type solitons.

Findings

Complete conformal gradient solitons are either product spaces, Euclidean space, or spheres.

Any noncompact gradient Yamabe-type soliton with positive Ricci tensor is rotationally symmetric.

The results unify and extend previous classification results in geometric analysis.

Abstract

In this paper we prove that any complete conformal gradient soliton with nonnegative Ricci tensor is either isometric to a direct product $\mathbb{R}\times N^{n-1}$, or globally conformally equivalent to the Euclidean space $\mathbb{R}^{n}$ or to the round sphere $\mathbb{S}^{n}$. In particular, we show that any complete, noncompact, gradient Yamabe-type soliton with positive Ricci tensor is rotationally symmetric, whenever the potential function is nonconstant.