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Results in Mathematics

, 74:64 | Cite as

Translating Solitons for the Inverse Mean Curvature Flow

  • Daehwan Kim
  • Juncheol PyoEmail author
Article
  • 65 Downloads

Abstract

In this paper, we investigate translating solitons for the inverse mean curvature flow (IMCF), which is a special solution deformed only for translation under the flow. The IMCF has been studied extensively not only as a type of a natural geometric flow, but also for obtaining various interesting geometric inequalities. We show that the translating solitons that are either ruled surfaces or translation surfaces are cycloid cylinders, and completely classify 2-dimensional helicoidal translating solitons and the higher dimensional rotationally symmetric translating solitons using the phase-plane analysis. The surface foliated by circles, which is called a cyclic surface, is regarded in terms of being the translating soliton for the IMCF, and then it is a surface of revolution whose revolution axis is parallel to the translating direction. In particular, we extend the result to a higher dimension, namely, the n-dimensional translating soliton foliated by spheres lying on parallel hyperplanes in \(\mathbb {R}^{n+1}\) must be a rotationally symmetric hypersurface whose rotation axis is parallel to the translating direction.

Keywords

Inverse mean curvature flow translating solitons cycloid cylinder helicoidal surface rotationally symmetric hypersurface 

Mathematics Subject Classification

53C44 37C10 53A10 

Notes

Acknowledgements

The authors warmly thank the referees for their careful reading of the paper and their valuable suggestions to improve the paper. The second author was supported in part by the National Research Foundation of Korea (NRF2017R1E1A1A03070495 and NRF-2017R1A5A1015722).

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of MathematicsKorea Institute for Advanced StudySeoulRepublic of Korea
  2. 2.Department of MathematicsPusan National UniversityBusanRepublic of Korea

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