Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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A CLASS OF INVERSE CURVATURE FLOWS IN ℝn+1, II

  • Hu, Jin-Hua (Faculty of Mathematics and Statistics Key Laboratory of Applied Mathematics of Hubei Province Hubei University) ;
  • Mao, Jing (Faculty of Mathematics and Statistics Key Laboratory of Applied Mathematics of Hubei Province Hubei University) ;
  • Tu, Qiang (Faculty of Mathematics and Statistics Key Laboratory of Applied Mathematics of Hubei Province Hubei University) ;
  • Wu, Di (Faculty of Mathematics and Statistics Key Laboratory of Applied Mathematics of Hubei Province Hubei University)
  • Received : 2019.09.16
  • Accepted : 2020.01.09
  • Published : 2020.09.01
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Abstract

We consider closed, star-shaped, admissible hypersurfaces in ℝn+1 expanding along the flow Ẋ = |X|α-1 F-β, α ≤ 1, β > 0, and prove that for the case α ≤ 1, β > 0, α + β ≤ 2, this evolution exists for all the time and the evolving hypersurfaces converge smoothly to a round sphere after rescaling. Besides, for the case α ≤ 1, α + β > 2, if furthermore the initial closed hypersurface is strictly convex, then the strict convexity is preserved during the evolution process and the flow blows up at finite time.

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References

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