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Richard H. Bamler

Publications and source records attributed to Richard H. Bamler.

31 records · Page 2Linked to original sources
arXiv2017

Long-time behavior of 3 dimensional Ricci flow -- Introduction

In the following series of papers we analyze the long-time behavior of 3 dimensional Ricci flows with surgery. Our main result will be that if the surgeries are performed correctly, then only finitely many surgeries occur and after some time the curvature is bounded by $C t^{-1}$. This result confirms a conjecture of Perelman. In the course of the proof, we also obtain a qualitative description of the geometry as $t \to \infty$.

math.DG↗
arXiv2017

Long-time behavior of 3 dimensional Ricci flow -- A: Generalizations of Perelman's long-time estimates

This is the first of a series of papers on the long-time behavior of 3 dimensional Ricci flows with surgery. In this paper we first fix a notion of Ricci flows with surgery, which will be used in this and the following three papers. Then we review Perelman's long-time estimates and generalize them to the case in which the underlying manifold is allowed to have a boundary. Eventually, making use of Perelman's techniques, we prove new long-time estimates, which hold whenever the metric is sufficiently collapsed.

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arXiv2017

Long-time behavior of 3 dimensional Ricci flow -- B: Evolution of the minimal area of simplicial complexes under Ricci flow

In this second part of a series of papers on the long-time behavior of Ricci flows with surgery, we establish a bound on the evolution of the infimal area of simplicial complexes inside a 3-manifold under the Ricci flow. This estimate generalizes an area estimate of Hamilton, which we will recall in the first part of the paper. We remark that in this paper we will mostly be dealing with non-singular Ricci flows. The existence of surgeries will not play an important role.

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arXiv2017

Long-time behavior of 3 dimensional Ricci flow -- D: Proof of the main results

This is the fourth and last part of a series of papers on the long-time behavior of 3 dimensional Ricci flows with surgery. In this paper, we prove our main two results. The first result states that if the surgeries are performed correctly, then the flow becomes non-singular eventually and the curvature is bounded by $C t^{-1}$. The second result provides a qualitative description of the geometry as $t \to \infty$.

math.DG↗
arXiv2016

Structure theory of singular spaces

In this paper we generalize the theory of Cheeger, Colding and Naber to certain singular spaces that arise as limits of sequences of Riemannian manifolds. This theory will have applications in the analysis of Ricci flows of bounded curvature, which we will describe in a subsequent paper.

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arXiv2016

Compactness properties of Ricci flows with bounded scalar curvature

In this paper we prove a compactness result for Ricci flows with bounded scalar curvature and entropy. It states that given any sequence of such Ricci flows, we can pass to a subsequence that converges to a metric space which is smooth away from a set of codimension $\geq 4$. The result has two main consequences: First, it implies that singularities in Ricci flows with bounded scalar curvature have codimension $\geq 4$ and, second, it establishes a general form of the Hamilton-Tian Conjecture, which is even true in the Riemannian case. In the course of the proof, we will also establish the following results: $L^{p < 4}$ curvature bounds, integral bounds on the curvature radius, Gromov-Hausdorff closeness of time-slices, an $\varepsilon$-regularity theorem for Ricci flows and an improved backwards pseudolocality theorem.

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arXiv2015

Heat kernel and curvature bounds in Ricci flows with bounded scalar curvature

In this paper we analyze Ricci flows on which the scalar curvature is globally or locally bounded from above by a uniform or time-dependent constant. On such Ricci flows we establish a new time-derivative bound for solutions to the heat equation. Based on this bound, we solve several open problems: 1. distance distortion estimates, 2. the existence of a cutoff function, 3. Gaussian bounds for heat kernels, and, 4. a backward pseudolocality theorem, which states that a curvature bound at a later time implies a curvature bound at a slightly earlier time. Using the backward pseudolocality theorem, we next establish a uniform $L^2$ curvature bound in dimension 4 and we show that the flow in dimension 4 converges to an orbifold at a singularity. We also obtain a stronger $\varepsilon$-regularity theorem for Ricci flows. This result is particularly useful in the study of Kähler Ricci flows on Fano manifolds, where it can be used to derive certain convergence results.

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arXiv2015

Almost-rigidity and the extinction time of positively curved Ricci flows

We prove that Ricci flows with almost maximal extinction time must be nearly round, provided that they have positive isotropic curvature when crossed with $\mathbb{R}^{2}$. As an application, we show that positively curved metrics on $S^{3}$ and $RP^{3}$ with almost maximal width must be nearly round.

math.DG↗
arXiv2013

Long-time analysis of 3 dimensional Ricci flow III

In this paper we analyze the long-time behavior of 3 dimensional Ricci flows with surgery. Our main result is that if the surgeries are performed correctly, then only finitely many surgeries occur and after some time the curvature is bounded by $C t^{-1}$. This result confirms a conjecture of Perelman. In the course of the proof, we also obtain a qualitative description of the geometry as $t \to \infty$. This paper is the third part of a series. Previously, we had to impose a certain topological condition $\mathcal{T}_2$ to establish the finiteness of the surgeries and the curvature control. The objective of this paper is to remove this condition and to generalize the result to arbitrary closed 3-manifolds. This goal is achieved by a new area evolution estimate for minimal simplicial complexes, which is of independent interest.

math.DG↗
arXiv2012

Long-time analysis of 3 dimensional Ricci flow II

This is the second part of a series of papers analyzing the long-time behaviour of 3 dimensional Ricci flows with surgery. We generalize the methods developed in the first part and use them to treat cases in which the initial manifold satisfies a certain purely topological condition which is far more general than the one that we previously had to impose. Amongst others, we are able to treat initial topologies such as the 3-torus or $Σ\times S^1$ where $Σ$ is any surface of genus $\geq 1$. We prove that under this condition, only finitely many surgeries occur and that after some time the curvature is bounded by $C t^{-1}$. This partially answers an open question in Perelman's work, which was made more precise by Lott and Tian. In the process of the proof, we also find an interesting description of the geometry at large times, which even holds when the condition on the initial topology is violated. The methods presented in this paper will be refined to treat a more general case in a subsequent paper.

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arXiv2011

Long-time analysis of 3 dimensional Ricci flow I

In this paper we analyze the long-time behaviour of 3 dimensional Ricci flow with surgery. We prove that under the topological condition that the initial manifold only has non-aspherical or hyperbolic components in its geometric decomposition, there are only finitely many surgeries and the curvature is bounded by $C t^{-1}$ for large $t$. This answers an open question in Perelman's work, which was made more precise by Lott and Tian, for this class of initial topologies. More general classes will be discussed in subsequent papers using similar methods.

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