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.

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 On the mean curvature function for compact surfaces

1981 ◽  
Vol 16 (2) ◽  
pp. 179-183 ◽  
Author(s):  
H. Blaine Lawson, Jr. ◽  
Renato de Azevedo Tribuzy
Keyword(s):  
Mean Curvature ◽  
Curvature Function ◽  
The Mean ◽  
Compact Surfaces

ON THE MEAN CURVATURE FUNCTION FOR COMPACT SURFACES

1983 ◽  
pp. 141-145
Author(s):  
H. BLAINE LAWSON ◽  
RENATO DE AZEVEDO TRIBUZY
Keyword(s):  
Mean Curvature ◽  
Curvature Function ◽  
The Mean ◽  
Compact Surfaces

scholarly journals THE MEAN CURVATURE EQUATION ON SEMIDIRECT PRODUCTS : HEIGHT ESTIMATES AND SCHERK-LIKE GRAPHS

2016 ◽  
Vol 101 (1) ◽  
pp. 118-144
Author(s):  
ÁLVARO KRÜGER RAMOS
Keyword(s):  
Mean Curvature ◽  
Smooth Curve ◽  
Quasilinear Elliptic Equations ◽  
Curvature Function ◽  
Minimal Graph ◽  
Semidirect Products ◽  
Height Estimates ◽  
Mean Curvature Equation ◽  
The Mean ◽  
Quasilinear Elliptic

In the ambient space of a semidirect product $\mathbb{R}^{2}\rtimes _{A}\mathbb{R}$, we consider a connected domain ${\rm\Omega}\subseteq \mathbb{R}^{2}\rtimes _{A}\{0\}$. Given a function $u:{\rm\Omega}\rightarrow \mathbb{R}$, its ${\it\pi}$-graph is $\text{graph}(u)=\{(x,y,u(x,y))\mid (x,y,0)\in {\rm\Omega}\}$. In this paper we study the partial differential equation that $u$ must satisfy so that $\text{graph}(u)$ has prescribed mean curvature $H$. Using techniques from quasilinear elliptic equations we prove that if a ${\it\pi}$-graph has a nonnegative mean curvature function, then it satisfies some uniform height estimates that depend on ${\rm\Omega}$ and on the supremum the function attains on the boundary of ${\rm\Omega}$. When $\text{trace}(A)>0$, we prove that the oscillation of a minimal graph, assuming the same constant value $n$ along the boundary, tends to zero when $n\rightarrow +\infty$ and goes to $+\infty$ if $n\rightarrow -\infty$. Furthermore, we use these estimates, allied with techniques from Killing graphs, to prove the existence of minimal ${\it\pi}$-graphs assuming the value zero along a piecewise smooth curve ${\it\gamma}$ with endpoints $p_{1},\,p_{2}$ and having as boundary ${\it\gamma}\cup (\{p_{1}\}\times [0,\,+\infty ))\cup (\{p_{2}\}\times [0,\,+\infty ))$.

scholarly journals COMPLETE SPACELIKE SUBMANIFOLDS IN DE SITTER SPACES WITH

2013 ◽  
Vol 87 (3) ◽  
pp. 386-399 ◽  
Author(s):  
JIANCHENG LIU ◽  
JINGJING ZHANG
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Linear Relation ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
De Sitter ◽  
Spacelike Submanifolds ◽  
The Mean

AbstractIn this paper, we give a classification of spacelike submanifolds with parallel normalised mean curvature vector field and linear relation$R= aH+ b$of the normalised scalar curvature$R$and the mean curvature$H$in the de Sitter space${ S}_{p}^{n+ p} (c)$.

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.

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