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On the weakly conical expanding gradient Ricci solitons

Pak-Yeung Chan, Man-Chun Lee

TL;DR

The paper constructs new examples of complete expanding gradient Ricci solitons with positive curvature that defy quadratic decay expectations. By smoothing singular metrics on spheres and applying Deruelle's perturbation method together with a localized maximum principle, the authors produce AC expanders whose curvature either decays slowly (yet yields infinite $ASCR$) or remains non-decaying with cones possessing singularities. They demonstrate that the asymptotic scalar curvature ratio $ASCR(g)$ can be infinite in dimensions $n\ge 3$, providing counterexamples to proposed finite-decay conjectures, and showcase a variety of asymptotic geometries via Reifenberg and non-Reifenberg limits. The results reveal a rich landscape of curvature behavior for expanding solitons and highlight the roles of sphere-link smoothing, cone asymptotics, and compactness arguments in constructing and understanding these geometric objects.

Abstract

In this work, we construct several sequences of metrics on sphere with different limiting behaviors. By combining with the work of Deruelle, we use it and the localized maximum principle to construct various examples of expanding gradient Ricci solitons with positive curvature and exotic curvature decay. This answers a question proposed by Chow-Lu-Ni and also a question by Cao-Liu, respectively.

On the weakly conical expanding gradient Ricci solitons

TL;DR

The paper constructs new examples of complete expanding gradient Ricci solitons with positive curvature that defy quadratic decay expectations. By smoothing singular metrics on spheres and applying Deruelle's perturbation method together with a localized maximum principle, the authors produce AC expanders whose curvature either decays slowly (yet yields infinite ) or remains non-decaying with cones possessing singularities. They demonstrate that the asymptotic scalar curvature ratio can be infinite in dimensions , providing counterexamples to proposed finite-decay conjectures, and showcase a variety of asymptotic geometries via Reifenberg and non-Reifenberg limits. The results reveal a rich landscape of curvature behavior for expanding solitons and highlight the roles of sphere-link smoothing, cone asymptotics, and compactness arguments in constructing and understanding these geometric objects.

Abstract

In this work, we construct several sequences of metrics on sphere with different limiting behaviors. By combining with the work of Deruelle, we use it and the localized maximum principle to construct various examples of expanding gradient Ricci solitons with positive curvature and exotic curvature decay. This answers a question proposed by Chow-Lu-Ni and also a question by Cao-Liu, respectively.

Paper Structure

This paper contains 7 sections, 13 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Maximum principle along Ricci flows
  4. singular metrics on $\mathbb{S}^m$ and its smoothing
  5. Smoothing a Reifenburg metric
  6. Smoothing a non-Reifenburg metric
  7. Construction of solitons

Key Result

Theorem 1.1

Let $n\ge 3$. There exist infinitely many asymptotically conical (in the distance sense) complete expanding gradient Ricci solitons $(M^n,g, f)$ satisfying the following In particular, these examples have infinite scalar curvature ratio.

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 3.1: Theorem 1.1 in LeeTam2022
  • Remark 3.2
  • Theorem 4.1: Deruelle2016
  • Lemma 4.2
  • Proposition 4.3
  • proof
  • ...and 15 more