Bulletin of the Korean Mathematical Society (대한수학회보)

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STABILITY AND TOPOLOGY OF TRANSLATING SOLITONS FOR THE MEAN CURVATURE FLOW WITH THE SMALL Lm NORM OF THE SECOND FUNDAMENTAL FORM

  • Eungmo, Nam (Department of Mathematics Pusan National University) ;
  • Juncheol, Pyo (Department of Mathematics Pusan National University and School of Mathematics Korea Institute for Advanced Study)
  • Received : 2022.01.18
  • Accepted : 2022.07.15
  • Published : 2023.01.31
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Abstract

In this paper, we show that a complete translating soliton Σm in ℝn for the mean curvature flow is stable with respect to weighted volume functional if Σ satisfies that the Lm norm of the second fundamental form is smaller than an explicit constant that depends only on the dimension of Σ and the Sobolev constant provided in Michael and Simon [12]. Under the same assumption, we also prove that under this upper bound, there is no non-trivial f-harmonic 1-form of L2f on Σ. With the additional assumption that Σ is contained in an upper half-space with respect to the translating direction then it has only one end.

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Acknowledgement

The first author was supported in part by the National Research Foundation of Korea (NRF-2020R1A2C1A01005698) and the second author was supported in part by the National Research Foundation of Korea (NRF-2021R1A4A1032418).

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