• Title, Summary, Keyword: strictly convex hypersurface

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A CHARACTERIZATION OF ELLIPTIC HYPERBOLOIDS

  • Kim, Dong-Soo;Son, Booseon
    • Honam Mathematical Journal
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    • v.35 no.1
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    • pp.37-49
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    • 2013
  • Consider a non-degenerate open convex cone C with vertex the origin in the $n$2-dimensional Euclidean space $E^n$. We study volume properties of strictly convex hypersurfaces in the cone C. As a result, for example, if the volume of the region of an elliptic cone C cut off by the tangent hyperplane P of M at $p$ is independent of the point $p{\in}M$, then it is shown that the hypersurface M is part of an elliptic hyperboloid.

ROLLING STONES WITH NONCONVEX SIDES II: ALL TIME REGULARITY OF INTERFACE AND SURFACE

  • Lee, Ki-Ahm;Rhee, Eun-Jai
    • Journal of the Korean Mathematical Society
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    • v.49 no.3
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    • pp.585-604
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    • 2012
  • In this paper we consider the evolution of the rolling stone with a rotationally symmetric nonconvex compact initial surface ${\Sigma}_0$ under the Gauss curvature flow. Let $X:S^n{\times}[0,\;{\infty}){\rightarrow}\mathbb{R}^{n+1}$ be the embeddings of the sphere in $\mathbb{R}^{n+1}$ such that $\Sigma(t)=X(S^n,t)$ is the surface at time t and ${\Sigma}(0)={\Sigma}_0$. As a consequence the parabolic equation describing the motion of the hypersurface becomes degenerate on the interface separating the nonconvex part from the strictly convex side, since one of the curvature will be zero on the interface. By expressing the strictly convex part of the surface near the interface as a graph of a function $z=f(r,t)$ and the non-convex part of the surface near the interface as a graph of a function $z={\varphi}(r)$, we show that if at time $t=0$, $g=\frac{1}{n}f^{n-1}_{r}$ vanishes linearly at the interface, the $g(r,t)$ will become smooth up to the interface for long time before focusing.

A CLASS OF INVERSE CURVATURE FLOWS IN ℝn+1, II

  • Hu, Jin-Hua;Mao, Jing;Tu, Qiang;Wu, Di
    • Journal of the Korean Mathematical Society
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    • v.57 no.5
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    • pp.1299-1322
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    • 2020
  • 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.