Classification of noncollapsed translators in $\mathbb{R}^4$

Choi, Kyeongsu; Haslhofer, Robert; Hershkovits, Or

doi:10.48550/arxiv.2105.13819

Cited by 2 publications

(5 citation statements)

References 45 publications

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“…Hence, assume that the tangent flow at −∞ is a bubble-sheet, specifically, lim λ→0 λM λ −2 t = R 2 × S 1 ( 2|t|). The classification of complete noncompact ancient solutions under this assumption has been considered in [13] and [14]. The uniqueness of the one-parameter family of compact O(2)-symmetric ancient solutions constructed in [22] has been proved in [15].…”

mentioning

confidence: 99%

Unique asymptotics of ancient convex mean curvature flow solutions

Angenent

Daskalopoulos

Šešum

2019

J. Differential Geom.

124

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We study the compact noncollapsed ancient convex solutions to Mean Curvature Flow in R n+1 with O(1) × O(n) symmetry. We show they all have unique asymptotics as t → −∞ and we give precise asymptotic description of these solutions. In particular, solutions constructed by White, and Haslhofer and Hershkovits have those asymptotics (in the case of those particular solutions the asymptotics was predicted and formally computed by Angenent [2]).

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mentioning

confidence: 99%

Unique asymptotics of ancient convex mean curvature flow solutions

Angenent

Daskalopoulos

Šešum

2019

J. Differential Geom.

124

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“…In Section 5, we prove Theorem 1.3 (half-degenerate case). Specifically, in case the flow splits off a line, then using the classification from [BC19, ADS19, ADS20] we show that it must be R×2d-oval, and in case the flow is selfsimilarly translating, then using the classification from [CHH21b] we show that it belongs to the one-parameter family of 3d oval-bowls.…”

Section: Andmentioning

confidence: 99%

“…Consider now the case that our flow is selfsimilarly translating. Then, by the recent classification by Choi, Hershkovits and the second author [CHH21b] it is either R×2d-bowl, or a 3d round bowl, or belongs to the oneparameter family of 3d oval-bowls constructed by Hoffman-Ilmanen-Martin-White. The case of the 3d round bowl is excluded by the assumption that the tangent flow at −∞ is a bubble-sheet, and the case R×2d-bowl is excluded by the assumption rk(Q) = 1.…”

Section: The Half-degenerate Casementioning

confidence: 99%

“…1 Very recently, Choi, Hershkovits and the second author made significant progress towards the classification of ancient noncollapsed flows in R 4 . Specifically, it has been shown in [CHH21a] that there do not exist any winglike ancient noncollapsed flows, and it has been shown in [CHH21b] that any selfsimilarly translating solution is either R×2d-bowl, or a 3d round bowl, or 1 There is a parallel story for the Ricci flow. Namely, a similar classification for ancient κ-noncollapsed flows in 3d Ricci flow, as conjectured by Perelman, has been obtained in [Bre20,ABDS19,BDS20], with an extension to higher dimensions under the additional PIC2 assumption in [LZ18, BN20,BDNS21], but the classification of general ancient κnoncollapsed Ricci flows in higher dimensions has remained widely open.…”

Section: Introductionmentioning

confidence: 99%

“…For necks and certain bubble-sheets related sharp asymptotics have been upgraded to classification results in [ADS20, DH21,CHH21b]. In light of these prior results it seems likely that our sharp asymptotics from Theorem 1.4 (non-degenerate case) can be upgraded to show that any ancient noncollapsed flow in R 4 with rk(Q) = 2 is either the O(2)×O(2)-symmetric ancient oval from [HH16] or belongs to the one-parameter family of Z 2 × O(2)symmetric ovals from [DH21].…”

Section: Introductionmentioning

confidence: 99%

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Hearing the shape of ancient noncollapsed flows in $\mathbb{R}^{4}$

Du,

Haslhofer

2021

Preprint

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We consider ancient noncollapsed mean curvature flows in R 4 whose tangent flow at −∞ is a bubble-sheet. We carry out a fine spectral analysis for the bubble-sheet function u that measures the deviation of the renormalized flow from the round cylinder R 2 × S 1 ( √ 2) and prove that for τ → −∞ we have the fine asymptotics u(y, θ, τis a symmetric 2 × 2matrix whose eigenvalues are quantized to be either 0 or −1/ √ 8. This naturally breaks up the classification problem for general ancient noncollapsed flows in R 4 into three cases depending on the rank of Q. In the case rk(Q) = 0, generalizing a prior result of Choi, Hershkovits and the second author, we prove that the flow is either a round shrinking cylinder or R×2d-bowl. In the case rk(Q) = 1, under the additional assumption that the flow either splits off a line or is selfsimilarly translating, as a consequence of recent work by Angenent, Brendle, Choi, Daskalopoulos, Hershkovits, Sesum and the second author we show that the flow must be R×2d-oval or belongs to the one-parameter family of 3d oval-bowls constructed by Hoffman-Ilmanen-Martin-White, respectively. Finally, in the case rk(Q) = 2 we show that the flow is compact and SO(2)-symmetric and for τ → −∞ has the same sharp asymptotics as the O(2) × O(2)-symmetric ancient ovals constructed by Hershkovits and the second author. The full classification problem will be addressed in subsequent papers based on the results of the present paper. Contents 1. Introduction 1.1. Main results 1.2. Outline of the proofs 2. Fine bubble-sheet analysis and almost symmetry 2.1. Basic bubble-sheet setup 2.2. Graphical radius 2.3. Almost symmetry 2.4. Evolution expansion 3. Proof of the bubble-sheet quantization theorem 1 2 WENKUI DU, ROBERT HASLHOFER 3.1. Derivation of the spectral ODEs 18 3.2. Quantized asymptotics of the spectral ODEs 21 4. The fully-degenerate case 26

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Classification of noncollapsed translators in $\mathbb{R}^4$

Cited by 2 publications

References 45 publications

Unique asymptotics of ancient convex mean curvature flow solutions

Unique asymptotics of ancient convex mean curvature flow solutions

Hearing the shape of ancient noncollapsed flows in $\mathbb{R}^{4}$

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