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

2013 ◽  
Vol 50 (6) ◽  
pp. 1311-1332 ◽  
Author(s):  
Shunzi Guo ◽  
Guanghan Li ◽  
Chuanxi Wu
Keyword(s):  
Hyperbolic Space ◽  
Mean Curvature ◽  
The Mean ◽  
Convex Hypersurfaces

Horospherical convex hypersurfaces contracting of the hyperbolic space by functions of the mean curvature

2019 ◽  
Vol 30 (08) ◽  
pp. 1950039
Author(s):  
Shunzi Guo
Keyword(s):  
Euclidean Space ◽  
Hyperbolic Space ◽  
Mean Curvature ◽  
Single Point ◽  
Normal Vector ◽  
Unit Normal Vector ◽  
Final Time ◽  
Maximal Interval ◽  
The Mean ◽  
Convex Hypersurfaces

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 smooth function depending only on the mean curvature, and satisfies some further restrictions, without requiring homogeneity. 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. This generalizes the previous result [S. Guo, Convex hypersurfaces evolving by functions of the mean curvature, preprint (2016), arXiv:1610.08214 ] for convex hypersurfaces in the Euclidean space by the author to the setting in the hyperbolic space for the same class of flows.

scholarly journals Isometric immersions in the hyperbolic space with their image contained in a horoball

2001 ◽  
Vol 43 (1) ◽  
pp. 1-8 ◽  
Author(s):  
Ana Lluch
Keyword(s):  
Hyperbolic Space ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Complete Riemannian Manifold ◽  
Isometric Immersions ◽  
Special Groups ◽  
The Mean ◽  
Complex Hyperbolic Spaces ◽  
Real Hyperbolic Space ◽  
Sharp Lower Bound

We give a sharp lower bound for the supremum of the norm of the mean curvature of an isometric immersion of a complete Riemannian manifold with scalar curvature bounded from below into a horoball of a complex or real hyperbolic space. We also characterize the horospheres of the real or complex hyperbolic spaces as the only isometrically immersed hypersurfaces which are between two parallel horospheres, have the norm of the mean curvature vector bounded by the above sharp bound and have some special groups of symmetries.

scholarly journals QUANTITATIVE OSCILLATION ESTIMATES FOR ALMOST-UMBILICAL CLOSED HYPERSURFACES IN EUCLIDEAN SPACE

2015 ◽  
Vol 92 (1) ◽  
pp. 133-144 ◽  
Author(s):  
JULIAN SCHEUER
Keyword(s):  
Euclidean Space ◽  
Fundamental Form ◽  
Mean Curvature ◽  
Second Fundamental Form ◽  
Uniformly Convex ◽  
Explicit Dependence ◽  
Closed Hypersurfaces ◽  
Bounded Volume ◽  
The Mean ◽  
Convex Hypersurfaces

We prove${\it\epsilon}$-closeness of hypersurfaces to a sphere in Euclidean space under the assumption that the traceless second fundamental form is${\it\delta}$-small compared to the mean curvature. We give the explicit dependence of${\it\delta}$on${\it\epsilon}$within the class of uniformly convex hypersurfaces with bounded volume.

scholarly journals Gradient estimates for the constant mean curvature equation in hyperbolic space

2019 ◽  
Vol 150 (6) ◽  
pp. 3216-3230
Author(s):  
Rafael López
Keyword(s):  
Dirichlet Problem ◽  
Hyperbolic Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Gradient Estimates ◽  
Maximum Principles ◽  
Convex Domains ◽  
Curvature Equation ◽  
Mean Curvature Equation ◽  
The Mean

AbstractWe establish gradient estimates for solutions to the Dirichlet problem for the constant mean curvature equation in hyperbolic space. We obtain these estimates on bounded strictly convex domains by using the maximum principles theory of Φ-functions of Payne and Philippin. These estimates are then employed to solve the Dirichlet problem when the mean curvature H satisfies H < 1 under suitable boundary conditions.

scholarly journals Graphs with constant mean curvature in the 3-hyperbolic space

2002 ◽  
Vol 74 (3) ◽  
pp. 371-377 ◽  
Author(s):  
PEDRO A. HINOJOSA
Keyword(s):  
Hyperbolic Space ◽  
Smooth Function ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Three Dimensional ◽  
Curvature Surface ◽  
Smooth Functions ◽  
Planar Domains ◽  
Constant Mean Curvature Surface ◽  
The Mean

In this work we will deal with disc type surfaces of constant mean curvature in the three dimensional hyperbolic space which are given as graphs of smooth functions over planar domains. From the various types of graphs that could be defined in the hyperbolic space we consider in particular the horizontal and the geodesic graphs. We proved that if the mean curvature is constant, then such graphs are equivalent in the following sense: suppose that M is a constant mean curvature surface in the 3-hyperbolic space such that M is a geodesic graph of a function rho that is zero at the boundary, then there exist a smooth function f that also vanishes at the boundary, such that M is a horizontal graph of f. Moreover, the reciprocal is also true.

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