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

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 $α=0$ and $β\neq0$ (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.

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Remark 2.2
• proof : Proof of Lemma \ref{['main']}
• Corollary 2.3
• Corollary 2.4
• Proposition 3.1
• proof
• Lemma 3.2