On the existence of translating solutions of mean curvature flow in slab regions

Bourni, Theodora; Langford, Mat; Tinaglia, Giuseppe

doi:10.2140/apde.2020.13.1051

Cited by 17 publications

(14 citation statements)

References 22 publications

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“…x < . (27) Let S ⊆ R be a ∆-wing translater (see [2] and [20]) such that it is the graph of a convex function u :…”

Section: Sup σ δmentioning

confidence: 99%

Simply connected translating solitons contained in slabs

Chini¹

2020

Geometric Flows

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In this work we show that 2-dimensional, simply connected, translating solitons of the mean curvature flow embedded in a slab of ℝ3 with entropy strictly less than 3 must be mean convex and thus, thanks to a result by Spruck and Xiao are convex. Recently, such 2-dimensional convex translating solitons have been completely classified, up to an ambient isometry, as vertical planes, (tilted) grim reaper cylinders, Δ-wings and bowl translater. These are all contained in a slab, except for the rotationally symmetric bowl translater. New examples by Ho man, Martín and White show that the bound on the entropy is necessary.

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“…x < . (27) Let S ⊆ R be a ∆-wing translater (see [2] and [20]) such that it is the graph of a convex function u :…”

Section: Sup σ δmentioning

confidence: 99%

Simply connected translating solitons contained in slabs

Chini¹

2020

Geometric Flows

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“…In the plane there is only one (up to isometries): it is the well-known grim reaper of equation y = − log(cos(x)). For higher dimension there is much more freedom: the cylinder generated by the grim reaper is the simplest example, but it can be also deformed in order to give a one-parameter family of tilted grim reaper cylinders: see [4] or [13] for an exhaustive description. Altschuler and Wu [3] found the only rotationally symmetric translator that is a complete graph, the so-called bowl solution.…”

Section: Introductionmentioning

confidence: 99%

“…Clutterbuck, Schnürer, and Schulze [8] completed the classification of the rotationally invariant translators describing a one-parameter family of surfaces with two ends, often called translating catenoids. Very recently Bourni, Langford, and Tinaglia [4] found a class of graphical not symmetric translators. The biggest achievement in this subject probably is the recent classification of graphical translators in R 3 due to Hoffman, Ilmanen, Martin, and White [13]: they proved that tilted grim reapers, the bowl solutions, and the examples of [4] are the only complete graphs in the Euclidean 3-space that are translators.…”

Section: Introductionmentioning

confidence: 99%

“…Very recently Bourni, Langford, and Tinaglia [4] found a class of graphical not symmetric translators. The biggest achievement in this subject probably is the recent classification of graphical translators in R 3 due to Hoffman, Ilmanen, Martin, and White [13]: they proved that tilted grim reapers, the bowl solutions, and the examples of [4] are the only complete graphs in the Euclidean 3-space that are translators. A general classification is far from being understood.…”

Section: Introductionmentioning

confidence: 99%

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Invariant Translators of the Heisenberg Group

Pipoli

2020

J Geom Anal

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We classify all the translating solitons to the mean curvature flow in the threedimensional Heisenberg group that are invariant under the action of some oneparameter group of isometries of the ambient manifold. We provide a complete classification for any canonical deformation of the standard Riemannian metric of the Heisenberg group. We highlight similarities and differences with the analogous Euclidean translators: we mention in particular that we describe the analogous of the tilted grim reaper cylinders, of the bowl solution and of translating catenoids, but some of them are not convex in contrast with a recent result of Spruck and Xiao (Complete translating solitons to the mean curvature flow in R 3 with non-negative mean curvature, arXiv:1703.01003v2, 2017) in the Euclidean space. Moreover we also prove some non-existence results. Finally we study the convergence of these surfaces as the ambient metric converges to the standard sub-Riemannian metric on the Heisenberg group.

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“…Under such flows, spheres contract homothetically and always give rise to ancient solutions. On the other hand, various nontrivial examples of compact ancient solutions have been constructed during the years, for example, . In particular, we recall the White ovaloids : they are convex solutions which, as

t \to - \infty

, become more and more eccentric, but sweep the whole space.…”

Section: Introductionmentioning

confidence: 99%

Strong spherical rigidity of ancient solutions of expansive curvature flows

Risa

Sinestrari

2019

Bull. London Math. Soc.

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We consider geometric flows of hypersurfaces expanding by a function of the extrinsic curvature and we show that the homothethic sphere is the unique solution of the flow which converges to a point at the initial time. The result does not require assumptions on the speed other than positivity and monotonicity and it is proved using a reflection argument. Our theorem shows that expanding flows exhibit stronger spherical rigidity, if compared with the classification results of ancient solutions in the contractive case.

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On the existence of translating solutions of mean curvature flow in slab regions

Cited by 17 publications

References 22 publications

Simply connected translating solitons contained in slabs

Simply connected translating solitons contained in slabs

Invariant Translators of the Heisenberg Group

Strong spherical rigidity of ancient solutions of expansive curvature flows

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