Existence and asymptotic behavior of helicoidal translating solitons of the mean curvature flow

Daehwan Kim; Juncheol Pyo

doi:10.3934/dcds.2018256

Article Contents

2018, Volume 38, Issue 11: 5897-5919. Doi: 10.3934/dcds.2018256

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Existence and asymptotic behavior of helicoidal translating solitons of the mean curvature flow

Daehwan Kim^1,2, and
Juncheol Pyo^1,2, ,

1.
Department of Mathematics, Pusan National University, Busan 46241, Republic of Korea
2.
School of Mathematics, Korea Institute for Advanced Study, Seoul 02455, Republic of Korea

* Corresponding author
* Corresponding author

Received: March 2018

Revised: April 2018

Published online: November 2018

This research was supported in part by the National Research Foundation of Korea (NRF-2017R1E1A1A03070495 and NRF-2017R1A5A1015722).

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Abstract

Translating soliton is a special solution for the mean curvature flow (MCF) and the parabolic rescaling model of type Ⅱ singularities for the MCF. By introducing an appropriate coordinate transformation, we first show that there exist complete helicoidal translating solitons for the MCF in $\mathbb{R}^{3}$ and we classify the profile curves and analyze their asymptotic behavior. We rediscover the helicoidal translating solitons for the MCF which are founded by Halldorsson [10]. Second, for the pinch zero we rediscover rotationally symmetric translating solitons in $\Bbb R^{n+1}$ and analyze the asymptotic behavior of the profile curves using a dynamical system. Clearly rotational hypersurfaces are foliated by spheres. We finally show that translating solitons foliated by spheres become rotationally symmetric translating solitons with the axis of revolution parallel to the translating direction. Hence, we obtain that any translating soliton foliated by spheres becomes either an n-dimensional translating paraboloid or a winglike translator.

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Mathematics Subject Classification: Primary: 53C44, 37C10, 53A10.

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Figure 1. Two integral curves of vector field V (h = 1)

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Figure 2. Two profile curves of vector field $V$ ($h = 1$)

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Figure 4. Profile curves (n = 2, 5)

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Figure 3. Two integral curves of vector field $V$ ($n = 2$)

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