scholarly journals Index estimates for surfaces with constant mean curvature in 3-dimensional manifolds

2020 ◽  
Vol 60 (1) ◽  
Author(s):  
Nicolau S. Aiex ◽  
Han Hong
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
3 Dimensional

scholarly journals Unitarization of monodromy representations and constant mean curvature trinoids in 3-dimensional space forms

2007 ◽  
Vol 75 (3) ◽  
pp. 563-581 ◽  
Author(s):  
N. Schmitt ◽  
M. Kilian ◽  
S.-P. Kobayashi ◽  
W. Rossman
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Dimensional Space ◽  
Space Forms ◽  
3 Dimensional

HELICOIDAL SURFACES WITH PRESCRIBED CURVATURES IN Nil3

2013 ◽  
Vol 24 (14) ◽  
pp. 1350107 ◽  
Author(s):  
DAE WON YOON ◽  
DONG-SOO KIM ◽  
YOUNG HO KIM ◽  
JAE WON LEE
Keyword(s):  
Heisenberg Group ◽  
Gaussian Curvature ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Smooth Functions ◽  
Constant Gaussian Curvature ◽  
3 Dimensional ◽  
Helicoidal Surfaces ◽  
Prescribed Gaussian Curvature

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.

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

2010 ◽  
Vol 07 (06) ◽  
pp. 961-978 ◽  
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.

4-Dimensional manifolds with nonnegative scalar curvature and CMC boundary

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.

scholarly journals On Bernstein–Heinz–Chern–Flanders inequalities

2008 ◽  
Vol 144 (2) ◽  
pp. 457-464 ◽  
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.

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

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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