Uniqueness of convex ancient solutions to mean curvature flow in $\mathbb{R}^3$

Brendle, Simon; Choi, Kyeongsu

doi:10.48550/arxiv.1711.00823

Cited by 10 publications

(35 citation statements)

References 26 publications

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“…This was improved by Haslhofer [Has15], who showed uniqueness of the bowl among translators that are convex and noncollapsed, and more recently by Hershkovits [Her18], who proved uniqueness among translators, not necessarily convex, that are asymptotic to a cylinder. Most closely related to the present article are the recent result by Brendle-Choi [BC17], which proves uniqueness of the bowl among noncompact ancient solutions that are noncollapsed and convex, and the recent result by Angenent-Daskalopoulos-Sesum [ADS18], which proves uniqueness of the ancient ovals among compact ancient solutions that are noncollapsed and convex. Other highly important results that play a direct role in the present paper are the classification of low entropy shrinkers by Bernstein-Wang [BW17], and the classification of genus zero shrinkers by Brendle [Bre16].…”

Section: Introductionsupporting

confidence: 56%

“…Let us recall some facts from [BC17] about the operator L defined in (71). In cylindrical coordinates this operator takes the form…”

Section: Fine Neck Analysismentioning

confidence: 99%

“…In fact, by a result of Colding-Minicozzi [CM16] the space-time singular set is contained in finitely many compact embedded Lipschitz curves together with a countable set of point singularities. Moreover, the recent work of Brendle-Choi [BC17] and Angenent-Daskalopoulos-Sesum [ADS18] provides a short list of all potential blowup limits (singularity models) in the flow of mean convex surfaces: the round shrinking sphere, the round shrinking cylinder, the translating bowl soliton [AW94], and the ancient ovals [Whi00,HH16].…”

Section: Introductionmentioning

confidence: 99%

“…To get started, in Section 4.1 we set up the fine neck analysis similarly as in Angenent-Daskalopoulos-Sesum [ADS15] and Brendle-Choi [BC17]. The analysis is governed by the linear operator (24) L = ∆ − 1 2 x tan • ∇ + 1 on the round cylinder.…”

Section: Introductionmentioning

confidence: 99%

“…uniformly in time. The proof is based on the Brendle-Choi neck improvement theorem [BC17] and some careful estimates controlling how the fine necks align with each other. In particular, the estimate uniformly bounds the motion of the tip in the xy-plane.…”

Section: Introductionmentioning

confidence: 99%

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Ancient low entropy flows, mean convex neighborhoods, and uniqueness

Choi¹,

Haslhofer²,

Hershkovits³

2018

Preprint

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In this article, we prove the mean convex neighborhood conjecture for the mean curvature flow of surfaces in R 3 . Namely, if the flow has a spherical or cylindrical singularity at a space-time point X = (x, t), then there exists a positive ε = ε(X) > 0 such that the flow is mean convex in a space-time neighborhood of size ε around X. The major difficulty is to promote the infinitesimal information about the singularity to a conclusion of macroscopic size. In fact, we prove a more general classification result for all ancient low entropy flows that arise as potential limit flows near X. Namely, we prove that any ancient, unit-regular, cyclic, integral Brakke flow in R 3 with entropy at most 2π/e + δ is either a flat plane, a round shrinking sphere, a round shrinking cylinder, a translating bowl soliton, or an ancient oval. As an application, we prove the uniqueness conjecture for mean curvature flow through spherical or cylindrical singularities. In particular, assuming Ilmanen's multiplicity one conjecture, we conclude that for embedded two-spheres the mean curvature flow through singularities is well-posed. 1 2 KYEONGSU CHOI, ROBERT HASLHOFER, OR HERSHKOVITS 5.2. Cap size control and asymptotics 51 6. Rotational symmetry 53 6.1. Fine expansion away from the cap 53 6.2. Moving plane method 57 7. Classification of ancient low entropy flows 61 7.1. The noncompact case 61 7.2. The compact case 64 8. Applications 65 8.1. Proof of the mean convex neighborhood conjecture 65 8.2. Proof of the uniqueness conjectures for weak flows 70 References 71

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Section: Introductionsupporting

confidence: 56%

“…Let us recall some facts from [BC17] about the operator L defined in (71). In cylindrical coordinates this operator takes the form…”

Section: Fine Neck Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Ancient low entropy flows, mean convex neighborhoods, and uniqueness

Choi¹,

Haslhofer²,

Hershkovits³

2018

Preprint

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Convex Ancient Solutions to Mean Curvature Flow

Bourni¹,

Langford²,

Tinaglia³

2020

Differential Geometry in the Large

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For each n ≥ 3, we construct an O(2) × O(n − 1)-invariant ancient Ricci flow on the n-sphere S n with positive curvature operator and bounded "girth". As time approaches minus infinity, it decomposes into a flat cylinder S 1 × R n−1 in the "waist" region and an (n − 2)-plane of cigar solitons around any point in the "tip" region (both of which are of the same scale). This solution appears to be new, even in dimension three.

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