Translating Solitons for the Inverse Mean Curvature Flow

Kim, Dae-Hwan; Pyo, Juncheol

doi:10.1007/s00025-019-0990-2

Cited by 11 publications

(5 citation statements)

References 28 publications

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“…They also introduced an Euclidean product with the cycloid and ℝ, called a canonical cycloid cylinder, and its suitable combinations of a rotation and a dilation, called cycloid cylinders, as translating solitons for IMCF. The present authors [19] gave the existence and classification of rotationally symmetric translating solitons in ℝ +1 and provided those of helicoidal translating solitons in ℝ 3 . In the same paper, we proved that any cyclic translating soliton in ℝ 3 must have rotational symmetry with the rotation axis that is parallel to the direction of translation under IMCF, and we showed that either ruled translating solitons or translation surfaces to be the translating solitons in ℝ 3 are cycloid cylinders.…”

Section: Introductionmentioning

confidence: 86%

Remarks on solitons for inverse mean curvature flow

Kim

Pyo

2020

Mathematische Nachrichten

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Homothetic and translating solitons for the inverse mean curvature flow (IMCF) in ℝ +1 are self-similar solutions deformed by only homothety and translation under IMCF, respectively. In this paper, we address some geometric properties of these solitons. To be specific, the incompleteness for the homothetic soliton of velocity with 0 < < 1 and for any translating soliton are proved. In addition, we show that the quantitative area growth for the homothetic solitons for ≥ 1 and the translating solitons. K E Y W O R D S homothetic soliton, inverse mean curvature flow, translating soliton M S C (2 0 1 0) 53A10, 53C44

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Section: Introductionmentioning

confidence: 86%

Remarks on solitons for inverse mean curvature flow

Kim

Pyo

2020

Mathematische Nachrichten

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“…For a translating soliton for the inverse mean curvature flow, Drugan, Lee, and Wheeler [5] gave a translating soliton in R 2 which is the cycloid generated by a circle with radius 1 4 and gave a tilted cycloid product as a translating soliton in R 3 . Kim and Pyo [8,9] showed the existence and classification of rotationally symmetric translating solitons in R n+1 and showed that there is no complete translating soliton for inverse mean curvature flow in R n+1 .…”

Section: Introductionmentioning

confidence: 99%

Graphical translating solitons for the inverse mean curvature flow and isoparametric functions

Fujii¹

2022

Preprint

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In this paper, we consider a translating soliton for the inverse mean curvature flow given as a graph of a function on a domain in a unit sphere whose level sets give isoparametric foliation. First, we show that such function is given as a composition of an isoparametric function on the unit sphere and a function which is given as a solution of a certain ordinary differential equation. Further, we analyze the shape of the graphs of the solutions of the ordinary differential equation. This analysis leads to the classification of the shape of such translating solitons for the inverse mean curvature flow.

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“…Mean curvature and inverse mean curvature flows have seen significant progress in the last decades, helping to bring insight to deep problems related to the theory of hypersurfaces (see, for instance, [7,8,12,23,24,25,26]). In particular, translating solitons to these flows have been widely studied (see [14,22,27] and references therein).…”

Section: Introductionmentioning

confidence: 99%

Generalized Elastic Translating Solitons

Pámpano¹

2022

Preprint

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We study translating soliton solutions to the flow by powers of the curvature of curves in the plane. We characterize these solitons as critical curves for functionals depending on the curvature. More precisely, translating solitons to the flow by powers of the curvature are shown to be generalized elastic curves. In particular, focusing on the curve shortening flow, we deduce a new variational characterization of the grim reaper curve.

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Translating Solitons for the Inverse Mean Curvature Flow

Cited by 11 publications

References 28 publications

Remarks on solitons for inverse mean curvature flow

Remarks on solitons for inverse mean curvature flow

Graphical translating solitons for the inverse mean curvature flow and isoparametric functions

Generalized Elastic Translating Solitons

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