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Complete quasi-Yamabe gradient solitons with bounded scalar curvature

Shun Maeta

TL;DR

The paper addresses the classification of complete, nontrivial quasi-Yamabe gradient solitons under scalar curvature bounds within the Yamabe flow framework. It leverages the structure theorem of Catino–Mastrolia–Monticelli–Rigoli to reduce the problem to warped-product models and then applies curvature and monotonicity analyses to derive rigidity results. The main contributions include (i) showing direct-product types are necessarily trivial, (ii) proving rotational symmetry for shrinking/steady solitons with $R>\lambda$, and (iii) providing explicit warped-product models for expanding/steady solitons with $R<\lambda$, thereby mapping the landscape of complete quasi-Yamabe gradient solitons under bounded curvature. These results deepen understanding of singularity models in the Yamabe flow and relate to rigidity phenomena and symmetry properties of solitons.

Abstract

In this paper, we classify complete, nontrivial shrinking and steady quasi-Yamabe gradient solitons whose scalar curvature is bounded below by the soliton constant. We also classify complete, nontrivial expanding and steady quasi-Yamabe gradient solitons whose scalar curvature is bounded above by the soliton constant.

Complete quasi-Yamabe gradient solitons with bounded scalar curvature

TL;DR

The paper addresses the classification of complete, nontrivial quasi-Yamabe gradient solitons under scalar curvature bounds within the Yamabe flow framework. It leverages the structure theorem of Catino–Mastrolia–Monticelli–Rigoli to reduce the problem to warped-product models and then applies curvature and monotonicity analyses to derive rigidity results. The main contributions include (i) showing direct-product types are necessarily trivial, (ii) proving rotational symmetry for shrinking/steady solitons with , and (iii) providing explicit warped-product models for expanding/steady solitons with , thereby mapping the landscape of complete quasi-Yamabe gradient solitons under bounded curvature. These results deepen understanding of singularity models in the Yamabe flow and relate to rigidity phenomena and symmetry properties of solitons.

Abstract

In this paper, we classify complete, nontrivial shrinking and steady quasi-Yamabe gradient solitons whose scalar curvature is bounded below by the soliton constant. We also classify complete, nontrivial expanding and steady quasi-Yamabe gradient solitons whose scalar curvature is bounded above by the soliton constant.
Paper Structure (5 sections, 8 theorems, 12 equations, 1 table)

This paper contains 5 sections, 8 theorems, 12 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Triviality and results for complete quasi-Yamabe gradient solitons of all types
  4. Shrinking and Steady quasi-Yamabe gradient solitons with $R>\lambda$
  5. Expanding and Steady quasi-Yamabe gradient solitons with $R<\lambda$

Key Result

Theorem 2.1

Let $(M,g,F,c)$ be an $n$-dimensional complete quasi-Yamabe gradient soliton. Then $(M,g,F,c)$ is either $(1)$$M=\mathbb{R}\times N$ and $g=dr^2+\psi^2(r)g_N$, and $F$ has no critical point, or $(2)$$M=[0,+\infty)\times \mathbb{S}^{n-1}$ and $g=dr^2+\psi^2(r)g_S,$$r\in[0,+\infty)$, and $F$ has only

Theorems & Definitions (14)

  • Theorem 2.1: CMMR17
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Theorem 4.1
  • ...and 4 more