Classification of generalized Yamabe solitons under vanishing conditions on the Weyl, Cotton, and Cao-Chen tensors

TL;DR

This paper classifies complete conformal gradient solitons, including Yamabe and related structures, under vanishing conditions on certain curvature tensors, revealing symmetry and geometric properties.

Contribution

It provides a classification of locally conformally flat conformal gradient solitons and explores symmetry under curvature conditions, extending Yamabe soliton theory.

Findings

Complete nontrivial locally conformally flat conformal gradient solitons are classified.

Nonflat solitons with nonnegative scalar curvature are rotationally symmetric.

Vanishing Cotton or Cao-Chen tensors lead to specific classifications.

Abstract

We study complete conformal gradient solitons, a class containing gradient Yamabe solitons and many generalized Yamabe-type structures, including gradient almost Yamabe, gradient k-Yamabe, and gradient h-almost Yamabe solitons, and, after a change of the potential function, gradient Einstein-type manifolds with $\alpha=0$ and $\beta\neq0$ (in particular, quasi-Yamabe solitons). In this paper, we classify complete nontrivial locally conformally flat conformal gradient solitons. This result contributes to an analogue of Perelman's conjecture for Yamabe-type solitons. Moreover, we show that under nonnegative scalar curvature, every nonflat soliton is rotationally symmetric. We also obtain classifications assuming the Cotton or Cao-Chen tensor vanishes.