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Complete minimal hypersurfaces in the hyperbolic space $\mathbb{H}^4$ with vanishing Gauss-Kronecker curvature  [PDF]
T. Hasanis,A. Savas-Halilaj,T. Vlachos
Mathematics , 2005,
Abstract: We investigate 3-dimensional complete minimal hypersurfaces in the hyperbolic space $\mathbb{H}^{4}$ with Gauss-Kronecker curvature identically zero. More precisely, we give a classification of complete minimal hypersurfaces with Gauss-Kronecker curvature identically zero, nowhere vanishing second fundamental form and scalar curvature bounded from below.
Real hypersurfaces in complex and quaternionic hyperbolic spaces  [PDF]
Thomas Murphy
Mathematics , 2010,
Abstract: We introduce curvature-adapted foliations of complex hyperbolic space and study some of their properties. Generalized pseudo-Einstein hypersurfaces of complex hyperbolic space are classified. Analogous results for curvature-adapted hypersurfaces of quaternionic hyperbolic space are also obtained.
Stable minimal hypersurfaces in the hyperbolic space  [PDF]
Keomkyo Seo
Mathematics , 2010,
Abstract: In this paper we give an upper bound of the first eigenvalue of the Laplace operator on a complete stable minimal hypersurface $M$ in the hyperbolic space which has finite $L^2$-norm of the second fundamental form on $M$. We provide some sufficient conditions for minimal hypersurface of the hyperbolic space to be stable. We also describe stability of catenoids and helicoids in the hyperbolic space. In particular, it is shown that there exists a family of stable higher-dimensional catenoids in the hyperbolic space.
Monotonicity formula for complete hypersurfaces in the Hyperbolic space and applications  [PDF]
Hilário Alencar,Gregório Silva Neto
Mathematics , 2015,
Abstract: In this paper we prove a monotonicity formula for the integral of the mean curvature for complete and proper hypersurfaces of the hyperbolic space and, as consequences, we obtain a lower bound for the integral of the mean curvature and that the integral of the mean curvature is infinity.
Hyperbolic Geometry and Potential Theory on Minimal Hypersurfaces  [PDF]
Joachim Lohkamp
Mathematics , 2015,
Abstract: This is the second in a series of papers where we estab- lish skin structural concepts and results for singular area minimizing hypersurfaces. Here we conformally unfold these spaces to complete Gromov hyperbolic spaces with bounded geometry and we recover their singular set as the Gromov boundary but also as the Martin boundary for many classical elliptic operators.
Curvature flow of complete hypersurfaces in hyperbolic space  [PDF]
Ling Xiao
Mathematics , 2011,
Abstract: In this paper we continue our study of finding the curvature flow of complete hypersurfaces in hyperbolic space with a prescribed asymptotic boundary at infinity. Our main results are proved by deriving a priori global gradient estimates and C^2 estimates.
Completeness of hyperbolic centroaffine hypersurfaces  [PDF]
Vicente Cortés,Marc Nardmann,Stefan Suhr
Mathematics , 2014,
Abstract: This paper is concerned with the completeness (with respect to the centroaffine metric) of hyperbolic centroaffine hypersurfaces which are closed in the ambient vector space. We show that completeness holds under generic regularity conditions on the boundary of the convex cone generated by the hypersurface. The main result is that completeness holds for hyperbolic components of level sets of homogeneous cubic polynomials. This implies that every such component defines a complete quaternionic K\"ahler manifold of negative scalar curvature.
Phase transitions and minimal hypersurfaces in hyperbolic space]  [PDF]
Adriano Pisante,Marcello Ponsiglione
Mathematics , 2010,
Abstract: The purpose of this paper is to investigate the Cahn-Hillard approximation for entire minimal hypersurfaces in the hyperbolic space. Combining comparison principles with minimization and blow-up arguments, we prove existence results for entire local minimizers with prescribed behaviour at infinity. Then, we study the limit as the length scale tends to zero through a $\Gamma$-convergence analysis, obtaining existence of entire minimal hypersurfaces with prescribed boundary at infinity. In particular, we recover some existence results proved in M. Anderson and U. Lang using geometric measure theory.
Complete minimal hypersurfaces of $S^4$ with zero Gauss-Kronecker curvature  [PDF]
T. Hasanis,A. Savas-Halilaj,T. Vlachos
Mathematics , 2004,
Abstract: We investigate the structure of 3-dimensional complete minimal hypersurfaces in the unit sphere with Gauss-Kronecker curvature identically zero.
Curvature flow of complete convex hypersurfaces in hyperbolic space  [PDF]
Ling Xiao
Mathematics , 2011,
Abstract: We investigate the existence, convergence and uniqueness of modified general curvature flow of convex hypersurfaces in hyperbolic space with a prescribed asymptotic boundary.
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