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PIC1 pinched manifolds are flat or compact

Alix Deruelle, Man-Chun Lee, Felix Schulze, Miles Simon, Peter M. Topping

Abstract

Hamilton's pinching conjecture, that three-dimensional complete non-compact manifolds with pinched Ricci curvature are flat, has recently been resolved using Ricci flow. In this paper we prove a direct analogue of that result in all dimensions. In order to do so we develop a lifting technique that allows us to handle manifolds that are collapsed at infinity. This new method also gives an alternative way of handling collapsed manifolds in the known three-dimensional case. As part of this approach, we prove a Ricci flow curvature estimate of a type that would normally be derived from the Harnack inequality, but without requiring the strong curvature positivity hypothesis demanded by Harnack. We give an improved gap theorem as a further application.

PIC1 pinched manifolds are flat or compact

Abstract

Hamilton's pinching conjecture, that three-dimensional complete non-compact manifolds with pinched Ricci curvature are flat, has recently been resolved using Ricci flow. In this paper we prove a direct analogue of that result in all dimensions. In order to do so we develop a lifting technique that allows us to handle manifolds that are collapsed at infinity. This new method also gives an alternative way of handling collapsed manifolds in the known three-dimensional case. As part of this approach, we prove a Ricci flow curvature estimate of a type that would normally be derived from the Harnack inequality, but without requiring the strong curvature positivity hypothesis demanded by Harnack. We give an improved gap theorem as a further application.
Paper Structure (11 sections, 18 theorems, 67 equations)

This paper contains 11 sections, 18 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Pinching theorems
  3. The gap phenomenon for manifolds with weakly PIC1
  4. Outline of the proof of Theorem \ref{['main_thm']}
  5. The PIC1 curvature condition
  6. Shrinking balls lemmas
  7. $\eta$-Ricci balls
  8. Lifted flow lemma
  9. Harnack conclusions with mild curvature hypotheses
  10. Proof of the main theorem \ref{['main_thm']}
  11. An improved gap theorem

Key Result

Theorem 1.1

Suppose $(M,g_0)$ is a closed Riemannian manifold that is PIC1. Then $M$ is diffeomorphic to a spherical space form.

Theorems & Definitions (31)

  • Theorem 1.1: Brendle brendlePIC1, PIC1 sphere theorem
  • Theorem 1.2: Hamilton's pinching conjecture, DeruelleSchulzeSimon2025, LeeTopping2022 and Lott2019
  • Theorem 1.3: Main theorem
  • Theorem 1.4: Brendle-Schoen BSsurvey
  • Theorem 1.5: Lee-Topping LeeTopping2022_PIC1
  • Theorem 1.6: Deruelle-Schulze-Simon DSS2
  • Theorem 1.7: Lee-Topping LeeTopping2022 and LeeTopping2022_PIC1
  • Remark 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 21 more