Constant mean curvature and totally umbilical biharmonic surfaces in 3-dimensional geometries

In the present paper, we study helicoidal surfaces in the 3-dimensional Heisenberg group Nil3. Also, we construct helicoidal surfaces in Nil3 with prescribed Gaussian curvature or mean curvature given by smooth functions. As the results, we classify helicoidal surfaces with constant Gaussian curvature or constant mean curvature.

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COMPLETE CMC SPACELIKE SURFACES WITH BOUNDED HYPERBOLIC ANGLE IN GENERALIZED ROBERTSON–WALKER SPACETIMES

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887810004658 ◽

2010 ◽

Vol 07 (06) ◽

pp. 961-978 ◽

Cited By ~ 9

Author(s):

MAGDALENA CABALLERO ◽

ALFONSO ROMERO ◽

RAFAEL M. RUBIO

Keyword(s):

Mean Curvature ◽

Constant Mean Curvature ◽

Existence Theorems ◽

Natural Extension ◽

Minkowski Spacetime ◽

Spacelike Surface ◽

3 Dimensional ◽

Spacelike Surfaces ◽

Hyperbolic Angle ◽

Wide Family

Complete spacelike surfaces with constant mean curvature (CMC) and bounded hyperbolic angle in Generalized Robertson–Walker (GRW) spacetimes, obeying certain natural curvature assumptions, are studied. This boundedness assumption arises as a natural extension of the notion of bounded hyperbolic image of a spacelike surface in the 3-dimensional Lorentz–Minkowski spacetime. The results obtained apply to complete CMC spacelike surfaces lying between two spacelike slices in an GRW spacetime, in the steady state spacetime and in a static GRW spacetime. As an application, uniqueness and non-existence theorems for certain CMC spacelike surface differential equations in a wide family of open GRW spacetimes are given.

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4-Dimensional manifolds with nonnegative scalar curvature and CMC boundary

Communications in Contemporary Mathematics ◽

10.1142/s0219199719500949 ◽

2020 ◽

pp. 1950094

Author(s):

Yaohua Wang

Keyword(s):

Scalar Curvature ◽

Upper Bound ◽

Mean Curvature ◽

Constant Mean Curvature ◽

Outer Boundary ◽

Compact Manifolds ◽

Manifolds With Boundary ◽

Flat Extension ◽

Asymptotically Flat ◽

3 Dimensional

In this paper, we will consider 4-dimensional manifolds with nonnegative scalar curvature and constant mean curvature (CMC) boundary. For compact manifolds with boundary, the influence of the nonnegativity of the region scalar curvature to the geometry of the boundary is considered. Some inequalities are established for manifolds with inner boundary and outer boundary. Even for compact manifolds without inner boundary, we can obtain some inequalities involving the geometric quantities of the boundary and give some obstruction. We also discuss the 4-dimensional asymptotically flat extension of the 3-dimensional Bartnik data with CMC boundary and provide the upper bound of the Bartnik mass.

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On Bernstein–Heinz–Chern–Flanders inequalities

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004107000643 ◽

2008 ◽

Vol 144 (2) ◽

pp. 457-464 ◽

Cited By ~ 4

Author(s):

J. L. M. BARBOSA ◽

G. P. BESSA ◽

J. F. MONTENEGRO

Keyword(s):

Lower Bounds ◽

Riemannian Manifolds ◽

Mean Curvature ◽

Constant Mean Curvature ◽

Point Of View ◽

Constant Mean Curvature Surfaces ◽

3 Dimensional ◽

Codimension One ◽

Simple Curves

AbstractWe give an interpretation of the Chern–Heinz inequalities for graphs in order to extend them to transversally oriented codimension one C2-foliations of Riemannian manifolds. It contains Salavessa's work on mean curvature of graphs and fully generalizes results of Barbosa–Kenmotsu–Oshikiri [3] and Barbosa–Gomes–Silveira [2] about foliations of 3-dimensional Riemannian manifolds by constant mean curvature surfaces. This point of view of the Chern–Heinz inequalities can be applied to prove a Haymann–Makai–Osserman inequality (lower bounds of the fundamental tones of bounded open subsets Ω ⊂ ℝ2 in terms of its inradius) for embedded tubular neighbourhoods of simple curves of ℝn.

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Minimal surfaces in euclidean 3-space and their mean curvature 1 cousins in hyperbolic 3-space

Anais da Academia Brasileira de Ciências ◽

10.1590/s0001-37652003000300002 ◽

2003 ◽

Vol 75 (3) ◽

pp. 271-278

Author(s):

Shoichi Fujimori

Keyword(s):

Minimal Surface ◽

Hyperbolic Space ◽

Minimal Surfaces ◽

Mean Curvature ◽

Constant Mean Curvature ◽

3 Dimensional

We show that the Hopf differentials of a pair of isometric cousin surfaces, a minimal surface in euclidean 3-space and a constant mean curvature (CMC) one surface in the 3-dimensional hyperbolic space, with properly embedded annular ends, extend holomorphically to each end. Using this result, we derive conditions for when the pair must be a plane and a horosphere.

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