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Rigidity of positively curved Steady gradient Ricci solitons on orbifolds

Yuxing Deng

TL;DR

The paper investigates rigidity phenomena for positively curved steady gradient Ricci solitons on orbifolds, establishing a scalar curvature lower bound framework and linking solitons to orbifold Ricci flow via a flow generated by the soliton potential. It proves two central rigidity results: (i) under κ-noncollapsed, positive curvature operator, compact singularity, and linear curvature decay, a steady GRS on an orbifold must be a finite Bryant soliton quotient; (ii) if the soliton is positively curved and asymptotically quotient cylindrical, it is also a Bryant soliton quotient. Together, these results extend Bryant soliton rigidity to orbifolds and illuminate the structure of singularity models in higher-dimensional Ricci flow on singular spaces.

Abstract

In this paper, we study gradient Ricci soitons on smooth orbifolds. We prove that the scalar curvature of a complete shrinking or steady gradient Ricci soliton on an orbifold is nonnegative. We also show that a complete $κ$-noncollapsed steady gradient Ricci soliton on a Riemannian orbifold with positive curvature operator, compact singularity and linear curvature decay must be a finite quotient of the Bryant soliton. Finally, we show that a complete steady gradient Ricci soliton on a Riemannian orbifold with positive sectional curvature must be a finite quotient of the Bryant soliton if it is asymptotically quotient cylindrical.

Rigidity of positively curved Steady gradient Ricci solitons on orbifolds

TL;DR

The paper investigates rigidity phenomena for positively curved steady gradient Ricci solitons on orbifolds, establishing a scalar curvature lower bound framework and linking solitons to orbifold Ricci flow via a flow generated by the soliton potential. It proves two central rigidity results: (i) under κ-noncollapsed, positive curvature operator, compact singularity, and linear curvature decay, a steady GRS on an orbifold must be a finite Bryant soliton quotient; (ii) if the soliton is positively curved and asymptotically quotient cylindrical, it is also a Bryant soliton quotient. Together, these results extend Bryant soliton rigidity to orbifolds and illuminate the structure of singularity models in higher-dimensional Ricci flow on singular spaces.

Abstract

In this paper, we study gradient Ricci soitons on smooth orbifolds. We prove that the scalar curvature of a complete shrinking or steady gradient Ricci soliton on an orbifold is nonnegative. We also show that a complete -noncollapsed steady gradient Ricci soliton on a Riemannian orbifold with positive curvature operator, compact singularity and linear curvature decay must be a finite quotient of the Bryant soliton. Finally, we show that a complete steady gradient Ricci soliton on a Riemannian orbifold with positive sectional curvature must be a finite quotient of the Bryant soliton if it is asymptotically quotient cylindrical.

Paper Structure

This paper contains 4 sections, 22 theorems, 144 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Basics of orbifolds
  3. The existence of corresponding Ricci flow
  4. Rigidity of positively curved steady GRS on orbifolds

Key Result

Theorem 1.2

Suppose $(\mathcal{M},g,f)$ is a complete gradient Ricci soliton on an $n$-dimensional Riemannian orbifold. If $\lambda\le0$, then $R(x)\ge 0$ for all $x\in M$. If $\lambda>0$, then $R\ge -C$ for some positive constant $C$.

Figures (1)

  • Figure 1: surgury on the soliton

Theorems & Definitions (53)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 43 more