scholarly journals Constant mean curvature spheres in homogeneous three-manifolds

2020 ◽  
Author(s):  
William H. Meeks ◽  
Pablo Mira ◽  
Joaquín Pérez ◽  
Antonio Ros
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Three Dimensional ◽  
Complete Classification ◽  
Homogeneous Manifolds ◽  
The Mean

Abstract We prove that two spheres of the same constant mean curvature in an arbitrary homogeneous three-manifold only differ by an ambient isometry, and we determine the values of the mean curvature for which such spheres exist. This gives a complete classification of immersed constant mean curvature spheres in three-dimensional homogeneous manifolds.

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.

scholarly journals SMYTH SURFACES AND THE DREHRISS

2005 ◽  
Vol 48 (3) ◽  
pp. 549-555
Author(s):  
John M. Burns ◽  
Michael J. Clancy
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Gauss Map ◽  
Immersed Surfaces ◽  
Trivial Group ◽  
The Mean ◽  
Isometric Deformations

AbstractIsometric deformations of immersed surfaces in Euclidean 3-space are studied by means of the drehriss. When the immersion is of constant mean curvature and the deformation preserves the mean curvature, we determine the drehriss explicitly in terms of the immersion and its Gauss map. These methods are applied to obtain an alternative classification of the Smyth surfaces, i.e. constant mean curvature immersions of the plane into Euclidean 3-space which admit the action of $S^1$ as a non-trivial group of internal isometries.

SOME CLASSIFICATIONS OF LORENTZIAN SURFACES WITH FINITE TYPE GAUSS MAP IN THE MINKOWSKI 4-SPACE

2015 ◽  
Vol 99 (3) ◽  
pp. 415-427 ◽  
Author(s):  
NURETTIN CENK TURGAY
Keyword(s):  
Minkowski Space ◽  
Finite Type ◽  
Minimal Surfaces ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Gauss Map ◽  
Complete Classification ◽  
Nonzero Constant ◽  
Dimensional Minkowski Space

In this paper we study the Lorentzian surfaces with finite type Gauss map in the four-dimensional Minkowski space. First, we obtain the complete classification of minimal surfaces with pointwise 1-type Gauss map. Then, we get a classification of Lorentzian surfaces with nonzero constant mean curvature and of finite type Gauss map. We also give some explicit examples.

scholarly journals A Class of Weingarten Surfaces in Euclidean 3-Space

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Yu Fu ◽  
Lan Li
Keyword(s):  
Mean Curvature ◽  
Complete Classification ◽  
Curvature Function ◽  
Weingarten Surfaces ◽  
The Mean

The class of biconservative surfaces in Euclidean 3-space𝔼3are defined in (Caddeo et al., 2012) by the equationA(grad H)=-H grad Hfor the mean curvature functionHand the Weingarten operatorA. In this paper, we consider the more general case that surfaces in𝔼3satisfyingA(grad H)=kH grad Hfor some constantkare called generalized bi-conservative surfaces. We show that this class of surfaces are linear Weingarten surfaces. We also give a complete classification of generalized bi-conservative surfaces in𝔼3.

Integral criteria for closed surfaces with constant mean curvature

2007 ◽  
Vol 463 (2081) ◽  
pp. 1199-1210
Author(s):  
Luca Guzzardi ◽  
Epifanio G Virga
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Second Variation ◽  
Closed Surfaces ◽  
Area Functional ◽  
Symmetric Surface ◽  
The Second Variation ◽  
Integral Identities

We propose three integral criteria that must be satisfied by all closed surfaces with constant mean curvature immersed in the three-dimensional Euclidean space. These criteria are integral identities that follow from requiring the second variation of the area functional to be invariant under rigid displacements. We obtain from them a new proof of the old result by Delaunay, to the effect that the sphere is the only closed axis-symmetric surface.

Classification of 3-dimensional left-invariant almost paracontact metric structures

2017 ◽  
Vol 17 (3) ◽  
Author(s):  
Giovanni Calvaruso ◽  
Antonella Perrone
Keyword(s):  
Lie Groups ◽  
Three Dimensional ◽  
Complete Classification ◽  
Natural Assumption ◽  
Geometric Properties ◽  
3 Dimensional ◽  
Metric Structures

AbstractWe study left-invariant almost paracontact metric structures on arbitrary three-dimensional Lorentzian Lie groups. We obtain a complete classification and description under a natural assumption, which includes relevant classes as normal and almost para-cosymplectic structures, and we investigate geometric properties of these structures.

Rotational λ – hypersurfaces in Euclidean spaces

2021 ◽  
Vol 30 (1) ◽  
pp. 29-40
Author(s):  
KADRI ARSLAN ◽  
ALIM SUTVEREN ◽  
BETUL BULCA
Keyword(s):  
Present Article ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Special Solution ◽  
Euclidean Spaces ◽  
Self Similar ◽  
The Mean ◽  
Similar Flows

Self-similar flows arise as special solution of the mean curvature flow that preserves the shape of the evolving submanifold. In addition, \lambda -hypersurfaces are the generalization of self-similar hypersurfaces. In the present article we consider \lambda -hypersurfaces in Euclidean spaces which are the generalization of self-shrinkers. We obtained some results related with rotational hypersurfaces in Euclidean 4-space \mathbb{R}^{4} to become self-shrinkers. Furthermore, we classify the general rotational \lambda -hypersurfaces with constant mean curvature. As an application, we give some examples of self-shrinkers and rotational \lambda -hypersurfaces in \mathbb{R}^{4}.

scholarly journals Isometric Immersions of Constant Mean Curvature and Triviality of the Normal Connection

1972 ◽  
Vol 45 ◽  
pp. 139-165 ◽  
Author(s):  
Joseph Erbacher
Keyword(s):  
Euclidean Space ◽  
Scalar Curvature ◽  
Mean Curvature ◽  
Additional Assumption ◽  
Constant Mean Curvature ◽  
Constant Scalar Curvature ◽  
Normal Connection ◽  
Isometric Immersions ◽  
The Mean ◽  
Negative Sectional Curvature

In a recent paper [2] Nomizu and Smyth have determined the hypersurfaces Mn of non-negative sectional curvature iso-metrically immersed in the Euclidean space Rn+1 or the sphere Sn+1 with constant mean curvature under the additional assumption that the scalar curvature of Mn is constant. This additional assumption is automatically satisfied if Mn is compact. In this paper we extend these results to codimension p isometric immersions. We determine the n-dimensional submanifolds Mn of non-negative sectional curvature isometrically immersed in the Euclidean Space Rn+P or the sphere Sn+P with constant mean curvature under the additional assumptions that Mn has constant scalar curvature and the curvature tensor of the connection in the normal bundle is zero. By constant mean curvature we mean that the mean curvature normal is paral lel with respect to the connection in the normal bundle. The assumption that Mn has constant scalar curvature is automatically satisfied if Mn is compact. The assumption on the normal connection is automatically sa tisfied if p = 2 and the mean curvature normal is not zero.

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