Complete translating solitons to the mean curvature flow in $\mathbb{R}^3$ with nonnegative mean curvature

Spruck, Joel; Xiao, Ling

doi:10.48550/arxiv.1703.01003

Cited by 21 publications

(38 citation statements)

References 28 publications

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“…Wang proved uniqueness of the bowl soliton among entire convex translating graphs [Wan11]. This was improved recently by Spruck-Xiao [SX17], who showed that every entire mean convex translating graph is in fact convex. 4 Without assuming that the translator is graphical, Martin, Savas-Halilaj and Smoczyk proved a uniqueness result for the bowl soliton under rather strong asymptotic assumptions [MSHS15].…”

Section: Introductionmentioning

confidence: 95%

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: Introductionmentioning

confidence: 95%

Ancient low entropy flows, mean convex neighborhoods, and uniqueness

Choi¹,

Haslhofer²,

Hershkovits³

2018

Preprint

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“…There exist strictly increasing sequences of positive numbers {m 1 i }, {m 2 i }, {n 1 i } and {n 2 i } so that m 1 i < m 2 i and −n 1 i < −n 2 i , for all i > i 0 ;; b. There exist smooth functions: (23) ϕ…”

Section: Thus By Theorem 23 We Getmentioning

confidence: 99%

“…He characterized the bowl soliton as the only convex translating soliton which is an entire graph. Very recently, J. Spruck and L. Xiao [23] have proved that a translating soliton which is graph over the whole R 2 must be convex. R. Haslhofer [9] showed that any strictly convex, uniformly two-convex translator which is non-collapsing is necessarily rotationally symmetric.…”

Section: Introductionmentioning

confidence: 99%

Translating solitons of the mean curvature flow asymptotic to hyperplanes in $\mathbb{R}^{n+1}$

Gama¹,

Martı́n²

2018

Preprint

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A translating soliton is a hypersurface M in R n+1 such that the family M t = M − t e n+1 is a mean curvature flow, i.e., such that normal component of the velocity at each point is equal to the mean curvature at that point H = e ⊥ n+1 . In this paper we obtain a characterization of hyperplanes which are parallel to the velocity and the family of tilted grim reaper cylinders as the only translating solitons in R n+1 which are C 1 -asymptotic to two half-hyperplanes outside a nonvertical cylinder. This result was proven for translators in R 3 by the second author, Perez-Garcia, Savas-Halilaj and Smoczyk under the additional hypotheses that the genus of the surface was locally bounded and the cylinder was perpendicular to the translating velocity.

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“…Further, Hoffman, Ilmanen, Martín and White [3] showed that no complete translating soliton is the graph of a function over a halfspace in R 2 . Spruck and Xiao [9] showed that complete translating soliton which is a graph over a domain in R 2 must be convex. Also, for the function u :…”

Section: Introductionmentioning

confidence: 99%

Graphical translating solitons for the mean curvature flow and isoparametric functions

Fujii¹

2021

Preprint

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In this paper, we consider a translating soliton for the mean curvature flow starting from a graph of a function on a domain in the n-dimensional unit sphere S n whose level sets are contained in isoparametric foliation. In this case, the function is given as a composition of an isoparametric function and a solution of a certain ordinary differential equation. We classify shapes of graphs of such solutions.

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Complete translating solitons to the mean curvature flow in $\mathbb{R}^3$ with nonnegative mean curvature

Cited by 21 publications

References 28 publications

Ancient low entropy flows, mean convex neighborhoods, and uniqueness

Ancient low entropy flows, mean convex neighborhoods, and uniqueness

Translating solitons of the mean curvature flow asymptotic to hyperplanes in $\mathbb{R}^{n+1}$

Graphical translating solitons for the mean curvature flow and isoparametric functions

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