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On the Hamilton-Lott conjecture in higher dimensions

Alix Deruelle, Felix Schulze, Miles Simon

Abstract

We study $n$-dimensional Ricci flows with non-negative Ricci curvature where the curvature is pointwise controlled by the scalar curvature and bounded by $C/t$, starting at metric cones which are Reifenberg outside the tip. We show that any such flow behaves like a self-similar solution up to an exponential error in time. As an application, we show that smooth $n$-dimensional complete non-compact Riemannian manifolds which are uniformly PIC1-pinched, with positive asymptotic volume ratio, are Euclidean. This confirms a higher dimensional version of a conjecture of Hamilton and Lott under the assumption of non-collapsing. It also yields a new and more direct proof of the original conjecture of Hamilton and Lott in three dimensions.

On the Hamilton-Lott conjecture in higher dimensions

Abstract

We study -dimensional Ricci flows with non-negative Ricci curvature where the curvature is pointwise controlled by the scalar curvature and bounded by , starting at metric cones which are Reifenberg outside the tip. We show that any such flow behaves like a self-similar solution up to an exponential error in time. As an application, we show that smooth -dimensional complete non-compact Riemannian manifolds which are uniformly PIC1-pinched, with positive asymptotic volume ratio, are Euclidean. This confirms a higher dimensional version of a conjecture of Hamilton and Lott under the assumption of non-collapsing. It also yields a new and more direct proof of the original conjecture of Hamilton and Lott in three dimensions.
Paper Structure (15 sections, 30 theorems, 126 equations)

This paper contains 15 sections, 30 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Outline of paper
  4. Acknowledgements
  5. Basics of Ricci flows with non-negative Ricci curvature
  6. Local elliptic regularizations of $d_0^2$
  7. Setup and local parabolic regularizations of $d_0^2$
  8. Equations
  9. Pointwise interior estimates
  10. Integral estimates
  11. Pointwise decay on the limit solution
  12. Ricci-pinched almost expanding gradient Ricci solitons
  13. Perelman cut-off functions
  14. Curvature constraints
  15. Sharp Poisson regularization of $d_0^2$

Key Result

Theorem 1.1

Let $(M^n,g(t))_{t\in(0,T)}$ be a smooth, complete, connected Ricci flow such that there exists $D_0 \in (0,\infty)$ such that on $M\times(0,T)$: Assume that the pointed limit in the distance sense of $(M^n,d_{g(t)},o)$ as $t$ goes to $0$ is a metric cone $(C(X),d_0,o)$ that is uniformly locally $n$-Reifenberg outside its tip $o$. Assume further that on $M\times(0,T)$ for some $D_1 \in (0,\infty)

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4: $Lee-Top-3d\text{, }Lee-Top-PIC2$
  • Theorem 1.5: $Lee-Top-PIC2$
  • Proposition 2.1
  • proof
  • Proposition 3.1
  • proof
  • Proposition 4.1
  • ...and 52 more