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Classification of low-dimensional complete gradient Yamabe solitons

Shun Maeta

TL;DR

The article achieves a complete classification of nontrivial non-flat three-dimensional complete gradient Yamabe solitons by applying Tashiro’s theorem, which forces solitons to be warped products with a one-dimensional base and a 2D fiber. It also bridges Ricci and Yamabe solitons in two dimensions to obtain a full classification of nontrivial complete gradient Ricci and Yamabe solitons in 2D, via a warped-product analysis of the potential function. For each soliton type (steady, shrinking, expanding) in 3D, the authors derive explicit warped-product models and, in the expanding case, provide new higher-dimensional examples. The work foregrounds expanding Yamabe solitons as rich sources of examples and clarifies how low-dimensional solitons fit within the broader Yamabe-Ricci soliton landscape, with implications for singularity models in the Yamabe flow.

Abstract

In this paper, we completely classify nontrivial non-flat three-dimensional complete gradient Yamabe solitons. Furthermore, by considering Ricci solitons from the point of view of Yamabe solitons, we provide a proof of the complete classification of nontrivial two-dimensional complete gradient Ricci solitons.

Classification of low-dimensional complete gradient Yamabe solitons

TL;DR

The article achieves a complete classification of nontrivial non-flat three-dimensional complete gradient Yamabe solitons by applying Tashiro’s theorem, which forces solitons to be warped products with a one-dimensional base and a 2D fiber. It also bridges Ricci and Yamabe solitons in two dimensions to obtain a full classification of nontrivial complete gradient Ricci and Yamabe solitons in 2D, via a warped-product analysis of the potential function. For each soliton type (steady, shrinking, expanding) in 3D, the authors derive explicit warped-product models and, in the expanding case, provide new higher-dimensional examples. The work foregrounds expanding Yamabe solitons as rich sources of examples and clarifies how low-dimensional solitons fit within the broader Yamabe-Ricci soliton landscape, with implications for singularity models in the Yamabe flow.

Abstract

In this paper, we completely classify nontrivial non-flat three-dimensional complete gradient Yamabe solitons. Furthermore, by considering Ricci solitons from the point of view of Yamabe solitons, we provide a proof of the complete classification of nontrivial two-dimensional complete gradient Ricci solitons.
Paper Structure (6 sections, 5 theorems, 23 equations)

This paper contains 6 sections, 5 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Classification of three-dimensional complete steady gradient Yamabe solitons
  4. Classification of three-dimensional complete shrinking gradient Yamabe solitons
  5. Classification of three-dimensional complete expanding gradient Yamabe solitons
  6. Classification of two-dimensional complete gradient Ricci solitons and Yamabe solitons

Key Result

Theorem 2.1

A complete Riemannian manifold $(M^n,g)$ which satisfies that for any smooth functions $F$ and $\varphi$ on $M$, $\nabla \nabla F=\varphi g$ is either $(1)$ compact and rotationally symmetric, or $(2)$ rotationally symmetric and equal to the warped product $([0,+\infty),dr^2)\times_{|\nabla F|}(\mat

Theorems & Definitions (13)

  • Remark 1.1
  • Theorem 2.1: Tashiro65
  • Remark 2.2
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof
  • Theorem 5.1
  • proof
  • Example
  • ...and 3 more