A constant mean curvature surface and the Dirac operator

1997 ◽  
Vol 30 (11) ◽  
pp. 4019-4029 ◽  
Author(s):  
Shigeki Matsutani
Keyword(s):  
Dirac Operator ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Curvature Surface ◽  
Constant Mean Curvature Surface

CHARACTERIZATIONS OF BIANCHI–BÄCKLUND TRANSFORMATIONS OF CONSTANT MEAN CURVATURE SURFACES

2005 ◽  
Vol 16 (02) ◽  
pp. 101-110 ◽  
Author(s):  
SHIMPEI KOBAYASHI ◽  
JUN-ICHI INOGUCHI
Keyword(s):  
Darboux Transformation ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Bäcklund Transformation ◽  
Bäcklund Transformations ◽  
Simple Type ◽  
Curvature Surface ◽  
Backlund Transformation ◽  
Constant Mean Curvature Surfaces ◽  
Constant Mean Curvature Surface

We show that Bianchi–Bäcklund transformation of a constant mean curvature surface is equivalent to the Darboux transformation and the simple type dressing.

scholarly journals Delaunay ends of constant mean curvature surfaces

2008 ◽  
Vol 144 (1) ◽  
pp. 186-220 ◽  
Author(s):  
M. Kilian ◽  
W. Rossman ◽  
N. Schmitt
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Weierstrass Representation ◽  
Curvature Surface ◽  
Regular Singularity ◽  
Constant Mean Curvature Surfaces ◽  
Constant Mean Curvature Surface ◽  
Delaunay Surface

AbstractThe generalized Weierstrass representation is used to analyze the asymptotic behavior of a constant mean curvature surface that arises locally from an ordinary differential equation (ODE) with a regular singularity. We prove that a holomorphic perturbation of an ODE that represents a Delaunay surface generates a constant mean curvature surface which has a properly immersed end that is asymptotically Delaunay. Furthermore, that end is embedded if the Delaunay surface is unduloidal.

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 Stability of a constant mean curvature surface in R3

1987 ◽  
Vol 36 (1) ◽  
pp. 19-24 ◽  
Author(s):  
Sung Eun Koh
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Curvature Surface ◽  
Nonzero Constant ◽  
Constant Mean Curvature Surface

We solve the question raised by Barbosa and do Carmo as to whether there exists a complete, noncompact stably immersed surface in R3 with nonzero constant mean curvature. We show that such a surface is necessarily minimal, that is, its mean curvature is zero.

scholarly journals THE FIRST EIGENVALUE OF THE DIRAC OPERATOR IN A CONFORMAL CLASS

2006 ◽  
Vol 03 (05n06) ◽  
pp. 833-844 ◽  
Author(s):  
BERND AMMANN ◽  
EMMANUEL HUMBERT
Keyword(s):  
Dirac Operator ◽  
Variational Problem ◽  
Mean Curvature ◽  
Unit Volume ◽  
Constant Mean Curvature ◽  
Positive Eigenvalue ◽  
First Eigenvalue ◽  
Conformal Class ◽  
Open Problems ◽  
The First Eigenvalue

In this overview article, we study the first positive eigenvalue of the Dirac operator in a unit volume conformal class. In particular, we discuss the question whether the infimum is attained. In the first part, we explain the corresponding variational problem. In the following parts we discuss the relation to the spinorial mass endomorphism and an application to surfaces of constant mean curvature. The article also mentions some open problems and work in progress.

scholarly journals TIME-INDEPENDENT SOLUTIONS TO THE TWO-DIMENSIONAL NONLINEAR O(3) SIGMA MODEL AND SURFACES OF CONSTANT MEAN CURVATURE

1995 ◽  
Vol 10 (03) ◽  
pp. 337-364 ◽  
Author(s):  
MICHAEL S. ODY ◽  
LEWIS H. RYDER
Keyword(s):  
Mean Curvature ◽  
Sigma Model ◽  
Constant Mean Curvature ◽  
Three Dimensional ◽  
Gauss Map ◽  
Field Vector ◽  
Dimensional Euclidean Space ◽  
Normal Vector ◽  
Constant Mean Curvature Surface ◽  
Spherical Images

It is shown that time-independent solutions to the (2+1)-dimensional nonlinear O(3) sigma model may be placed in correspondence with surfaces of constant mean curvature in three-dimensional Euclidean space. The tools required to establish this correspondence are provided by the classical differential geometry of surfaces. A constant-mean-curvature surface induces a solution to the O(3) model through the identification of the Gauss map, or normal vector, of the surface with the field vector of the sigma model. Some explicit solutions, including the solitons and antisolitons discovered by Belavin and Polyakov, and a more general solution due to Purkait and Ray, are considered and the surfaces giving rise to them are found explicitly. It is seen, for example, that the Belavin-Polyakov solutions are induced by the Gauss maps of surfaces which are conformal to their spherical images, i.e. spheres and minimal surfaces, and that the Purkait-Ray solution corresponds to the family of constant-mean-curvature helicoids first studied by do Carmo and Dajczer in 1982. A generalization of this method to include time dependence may shed new light on the role of the Hopf invariant in this model.

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