scholarly journals Darboux Transforms of a Harmonic Inverse Mean Curvature Surface

Geometry ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Katsuhiro Moriya
Keyword(s):  
Mean Curvature ◽  
Curvature Surface ◽  
Surface Of Revolution ◽  
Darboux Transforms

The notion of a generalized harmonic inverse mean curvature surface in the Euclidean four-space is introduced. A backward Bäcklund transform of a generalized harmonic inverse mean curvature surface is defined. A Darboux transform of a generalized harmonic inverse mean curvature surface is constructed by a backward Bäcklund transform. For a given isothermic harmonic inverse mean curvature surface, its classical Darboux transform is a harmonic inverse mean curvature surface. Then a transform of a solution to the Painlevé III equation in trigonometric form is defined by a classical Darboux transform of a harmonic inverse mean curvature surface of revolution.

scholarly journals Darboux transforms and simple factor dressing of constant mean curvature surfaces

2012 ◽  
Vol 140 (1-2) ◽  
pp. 213-236 ◽  
Author(s):  
F. E. Burstall ◽  
J. F. Dorfmeister ◽  
K. Leschke ◽  
A. C. Quintino
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Constant Mean Curvature Surfaces ◽  
Simple Factor ◽  
Darboux Transforms

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.

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