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Asymptotic Geometry of Four-Dimensional Steady Solitons

Aprameya Girish Hebbar, Natasa Sesum

Abstract

In this paper we study the behavior of the scalar curvature at infinity on complete noncompact steady gradient Ricci solitons. In dimension four, we assume that the canonical Ricci flow induced by the soliton is a weak $κ$-solution and that the soliton is not isometric to the Bryant soliton. In this setting, we identify the two edges of the soliton and prove that the scalar curvature decays at a linear rate away from these edges. Moreover, if the scalar curvature vanishes at infinity, then a stronger inequality holds and the asymptotic cone is a ray. In particular, our results apply to the four-dimensional flying wings constructed by Lai.

Asymptotic Geometry of Four-Dimensional Steady Solitons

Abstract

In this paper we study the behavior of the scalar curvature at infinity on complete noncompact steady gradient Ricci solitons. In dimension four, we assume that the canonical Ricci flow induced by the soliton is a weak -solution and that the soliton is not isometric to the Bryant soliton. In this setting, we identify the two edges of the soliton and prove that the scalar curvature decays at a linear rate away from these edges. Moreover, if the scalar curvature vanishes at infinity, then a stronger inequality holds and the asymptotic cone is a ray. In particular, our results apply to the four-dimensional flying wings constructed by Lai.
Paper Structure (12 sections, 34 theorems, 214 equations)

This paper contains 12 sections, 34 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. Setting
  3. Main Results
  4. Outline of the paper
  5. Acknowledgments
  6. Preliminaries
  7. Structure of level sets
  8. Stability of $(\epsilon,2)$-centers
  9. Scalar curvature along integral curves of $-\nabla f$
  10. Proof of the main result
  11. Asymptotic cone of the soliton
  12. Some auxiliary results

Key Result

Theorem 1.4

Let $(M^4,g,f)$ be a complete steady soliton satisfying assumption:A1, assumption:A2, assumption:A3, and assumption:A4. Then there exist two curves $\Gamma_1,\Gamma_2:[0,\infty)\to M$ with $\Gamma_i(0)=o$ such that $\Gamma_i((0,\infty))$ is an integral curve of $-\nabla f/|\nabla f|$, for $i=1,2$. I where $C$ depends on $\kappa$ and $(M,g)$.

Theorems & Definitions (88)

  • Conjecture 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • ...and 78 more