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Manifolds with PIC1 pinched curvature

Man-Chun Lee, Peter M. Topping

TL;DR

The paper addresses extending Hamilton's pinching results to higher dimensions by introducing a PIC1 pinching framework and proving a long-time Ricci-flow existence theory for complete non-compact manifolds with $R - \varepsilon_0\,\mathrm{scal}(R)\cdot I \in {\mathrm{C_{PIC1}}}$. The authors develop a cone-based approach, defining ${\mathrm{C_{PIC1}}}$, ${\mathrm{C_{PIC2}}}$ and the pinching families ${\hat{C}}(s)$, ${\check{C}}(s)$, and ${\tilde{C}}(b,s)$, and prove a local ODE-PDE preservation theorem that maintains pinching under Ricci flow. They prove local curvature decay estimates and curvature-improvement results, enabling a global Ricci-flow construction from PIC1-pinched initial data without assuming bounded curvature or non-collapsing. As a consequence, complete manifolds with non-negative complex sectional curvature and PIC1 pinching are shown to be flat or compact, with corollaries to spherical space-forms under stronger pinching, highlighting a significant rigidity phenomenon in higher dimensions.

Abstract

Recently it has been proved (Lee-Topping 2022, Deruelle-Schulze-Simon 2022, Lott 2019) that three-dimensional complete manifolds with non-negatively pinched Ricci curvature must be flat or compact, thus confirming a conjecture of Hamilton. In this paper we generalise our work on the existence of Ricci flows from non-compact pinched three-manifolds in order to prove a higher-dimensional analogue. We construct a solution to Ricci flow, for all time, starting with an arbitrary complete non-compact manifold that is PIC1 pinched. As an application we prove that any complete manifold of non-negative complex sectional curvature that is PIC1 pinched must be flat or compact.

Manifolds with PIC1 pinched curvature

TL;DR

The paper addresses extending Hamilton's pinching results to higher dimensions by introducing a PIC1 pinching framework and proving a long-time Ricci-flow existence theory for complete non-compact manifolds with . The authors develop a cone-based approach, defining , and the pinching families , , and , and prove a local ODE-PDE preservation theorem that maintains pinching under Ricci flow. They prove local curvature decay estimates and curvature-improvement results, enabling a global Ricci-flow construction from PIC1-pinched initial data without assuming bounded curvature or non-collapsing. As a consequence, complete manifolds with non-negative complex sectional curvature and PIC1 pinching are shown to be flat or compact, with corollaries to spherical space-forms under stronger pinching, highlighting a significant rigidity phenomenon in higher dimensions.

Abstract

Recently it has been proved (Lee-Topping 2022, Deruelle-Schulze-Simon 2022, Lott 2019) that three-dimensional complete manifolds with non-negatively pinched Ricci curvature must be flat or compact, thus confirming a conjecture of Hamilton. In this paper we generalise our work on the existence of Ricci flows from non-compact pinched three-manifolds in order to prove a higher-dimensional analogue. We construct a solution to Ricci flow, for all time, starting with an arbitrary complete non-compact manifold that is PIC1 pinched. As an application we prove that any complete manifold of non-negative complex sectional curvature that is PIC1 pinched must be flat or compact.
Paper Structure (5 sections, 22 theorems, 127 equations)

This paper contains 5 sections, 22 theorems, 127 equations.

Table of Contents

  1. introduction
  2. Pinching cones
  3. A priori estimates under pinching conditions
  4. Existence of Ricci flow under pinching
  5. A geometric interpretation of $\check{C} (s)$

Key Result

Theorem 1.1

Suppose $(M^3,g_0)$ is a complete (connected) three-dimensional Riemannian manifold with $\text{\rm Ric}\geq \varepsilon\, \mathrm{scal}\geq 0$ for some $\varepsilon>0$. Then $(M^3,g_0)$ is either flat or compact.

Theorems & Definitions (45)

  • Theorem 1.1: Hamilton's pinching conjecture, cf. CLN09
  • Theorem 1.2: Main theorem
  • Theorem 1.3
  • Remark 1.4
  • Corollary 1.5
  • Definition 2.1
  • Proposition 2.2: Propositions 13, 14 and 15 in BrendleSchoen2009, cf. BohmWilking2008
  • Proposition 2.3
  • proof
  • Lemma 3.1
  • ...and 35 more