scholarly journals Timelike surfaces with constant mean curvature in Lorentz three-space

2000 ◽  
Vol 52 (4) ◽  
pp. 515-532 ◽  
Author(s):  
Rafael López
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Three Space

scholarly journals Quaternionic representation of the moving frame for surfaces in Euclidean three-space and Lax pair

2004 ◽  
Vol 2004 (15) ◽  
pp. 755-762 ◽  
Author(s):  
Paul Bracken
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Lax Pair ◽  
Moving Frame ◽  
Lax Representation ◽  
Euclidean Three Space ◽  
Three Space ◽  
Codazzi Equations

The moving frame and associated Gauss-Codazzi equations for surfaces in three-space are introduced. A quaternionic representation is used to identify the Gauss-Weingarten equation with a particular Lax representation. Several examples are given, such as the case of constant mean curvature.

scholarly journals Delaunay surfaces expressed in terms of a Cartan moving frame

2020 ◽  
Vol 26 (1) ◽  
pp. 153-160
Author(s):  
Paul Bracken
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Elliptic Functions ◽  
Moving Frame ◽  
Jacobi Elliptic Functions ◽  
Surfaces Of Revolution ◽  
Constant Mean Curvature Surfaces ◽  
The Mean ◽  
Euclidean Three Space ◽  
Three Space

AbstractDelaunay surfaces are investigated by using a moving frame approach. These surfaces correspond to surfaces of revolution in the Euclidean three-space. A set of basic one-forms is defined. Moving frame equations can be formulated and studied. Related differential equations which depend on variables relevant to the surface are obtained. For the case of minimal and constant mean curvature surfaces, the coordinate functions can be calculated in closed form. In the case in which the mean curvature is constant, these functions can be expressed in terms of Jacobi elliptic functions.

scholarly journals On far-outlying constant mean curvature spheres in asymptotically flat Riemannian 3-manifolds

2020 ◽  
Vol 2020 (767) ◽  
pp. 161-191
Author(s):  
Otis Chodosh ◽  
Michael Eichmair
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Existence Results ◽  
Asymptotically Flat ◽  
Weingarten Surfaces ◽  
Differential Geom

AbstractWe extend the Lyapunov–Schmidt analysis of outlying stable constant mean curvature spheres in the work of S. Brendle and the second-named author [S. Brendle and M. Eichmair, Isoperimetric and Weingarten surfaces in the Schwarzschild manifold, J. Differential Geom. 94 2013, 3, 387–407] to the “far-off-center” regime and to include general Schwarzschild asymptotics. We obtain sharp existence and non-existence results for large stable constant mean curvature spheres that depend delicately on the behavior of scalar curvature at infinity.

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