• Title/Summary/Keyword: maximal hypersurfaces

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A NOTE ON MAXIMAL HYPERSURFACES IN A GENERALIZED ROBERTSON-WALKER SPACETIME

  • de Lima, Henrique Fernandes
    • Communications of the Korean Mathematical Society
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    • v.37 no.3
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    • pp.893-904
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    • 2022
  • In this note, we apply a maximum principle related to volume growth of a complete noncompact Riemannian manifold, which was recently obtained by Alías, Caminha and do Nascimento in [4], to establish new uniqueness and nonexistence results concerning maximal spacelike hypersurfaces immersed in a generalized Robertson-Walker (GRW) spacetime obeying the timelike convergence condition. A study of entire solutions for the maximal hypersurface equation in GRW spacetimes is also made and, in particular, a new Calabi-Bernstein type result is presented.

Maximal Hypersurfaces of (m + 2)-Dimensional Lorentzian Space Forms

  • Dursun, Ugur
    • Kyungpook Mathematical Journal
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    • v.48 no.1
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    • pp.109-121
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    • 2008
  • We determine maximal space-like hypersurfaces which are the images of subbundles of the normal bundle of m-dimensional totally geodesic space-like submanifolds of an (m + 2)-dimensional Lorentzian space form $\tilde{M}_1^{m+2}$(c) under the normal exponential map. Then we construct examples of maximal space-like hypersurfaces of $\tilde{M}_1^{m+2}$(c).

MAXIMAL SPACE-LIKE HYPERSURFACES IN H14(-1) WITH ZERO GAUSS-KRONECKER CURVATURE

  • CHENG QING-MING;SUH YOUNG JIN
    • Journal of the Korean Mathematical Society
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    • v.43 no.1
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    • pp.147-157
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    • 2006
  • In this paper, we study complete maximal space-like hypersurfaces with constant Gauss-Kronecker curvature in an antide Sitter space $H_1^4(-1)$. It is proved that complete maximal spacelike hypersurfaces with constant Gauss-Kronecker curvature in an anti-de Sitter space $H_1^4(-1)$ are isometric to the hyperbolic cylinder $H^2(c1){\times}H^1(c2)$ with S = 3 or they satisfy $S{\leq}2$, where S denotes the squared norm of the second fundamental form.

COMPLETE MAXIMAL SPACE-LIKE HYPERSURFACES IN AN ANTI-DE SITTER SPACE

  • Choi, Soon-Meen;Ki, U-Hang;Kim, He-Jin
    • Bulletin of the Korean Mathematical Society
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    • v.31 no.1
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    • pp.85-92
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    • 1994
  • It is well known that there exist no closed minimal surfaces in a 3-dimensional Euclidean space R$^{3}$. Myers [4] generalized the result to the case of the higher dimension and proved that there are no closed minimal hypersurfaces in an open hemisphere. The complete and non-compact version concerning Myers' theorem is recently considered by Cheng [1] and the following theorem is proved.

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DEFORMING PINCHED HYPERSURFACES OF THE HYPERBOLIC SPACE BY POWERS OF THE MEAN CURVATURE INTO SPHERES

  • Guo, Shunzi;Li, Guanghan;Wu, Chuanxi
    • Journal of the Korean Mathematical Society
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    • v.53 no.4
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    • pp.737-767
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    • 2016
  • This paper concerns closed hypersurfaces of dimension $n{\geq}2$ in the hyperbolic space ${\mathbb{H}}_{\kappa}^{n+1}$ of constant sectional curvature evolving in direction of its normal vector, where the speed equals a power ${\beta}{\geq}1$ of the mean curvature. The main result is that if the initial closed, weakly h-convex hypersurface satisfies that the ratio of the biggest and smallest principal curvature at everywhere is close enough to 1, depending only on n and ${\beta}$, then under the flow this is maintained, there exists a unique, smooth solution of the flow which converges to a single point in ${\mathbb{H}}_{\kappa}^{n+1}$ in a maximal finite time, and when rescaling appropriately, the evolving hypersurfaces exponential convergence to a unit geodesic sphere of ${\mathbb{H}}_{\kappa}^{n+1}$.

CONTRACTION OF HOROSPHERE-CONVEX HYPERSURFACES BY POWERS OF THE MEAN CURVATURE IN THE HYPERBOLIC SPACE

  • Guo, Shunzi;Li, Guanghan;Wu, Chuanxi
    • Journal of the Korean Mathematical Society
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    • v.50 no.6
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    • pp.1311-1332
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    • 2013
  • This paper concerns the evolution of a closed hypersurface of the hyperbolic space, convex by horospheres, in direction of its inner unit normal vector, where the speed equals a positive power ${\beta}$ of the positive mean curvature. It is shown that the flow exists on a finite maximal interval, convexity by horospheres is preserved and the hypersurfaces shrink down to a single point as the final time is approached.