The Asymptotic Shape of a Boundary Layer of Symmetric Willmore Surfaces of Revolution

2012 ◽  
pp. 19-29 ◽  
Author(s):  
Hans-Christoph Grunau
Keyword(s):  
Boundary Layer ◽  
Surfaces Of Revolution ◽  
Asymptotic Shape ◽  
Willmore Surfaces ◽  
Willmore Surfaces Of Revolution

Existence for Willmore surfaces of revolution satisfying non-symmetric Dirichlet boundary conditions

2019 ◽  
Vol 12 (4) ◽  
pp. 333-361 ◽  
Author(s):  
Sascha Eichmann ◽  
Hans-Christoph Grunau
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Lagrange Equation ◽  
Order Reduction ◽  
Dirichlet Boundary Conditions ◽  
Surfaces Of Revolution ◽  
Geometric Information ◽  
Willmore Surfaces ◽  
Willmore Surfaces Of Revolution ◽  
Willmore Energy

AbstractIn this paper, existence for Willmore surfaces of revolution is shown, which satisfy non-symmetric Dirichlet boundary conditions, if the infimum of the Willmore energy in the admissible class is strictly below {4\pi}. Under a more restrictive but still explicit geometric smallness condition we obtain a quite interesting additional geometric information: The profile curve of this solution can be parameterised as a graph over the x-axis. By working below the energy threshold of {4\pi} and reformulating the problem in the Poincaré half plane, compactness of a minimising sequence is guaranteed, of which the limit is indeed smooth. The last step consists of two main ingredients: We analyse the Euler–Lagrange equation by an order reduction argument by Langer and Singer and modify, when necessary, our solution with the help of suitable parts of catenoids and circles.

Willmore Surfaces of Revolution with Two Prescribed Boundary Circles

2011 ◽  
Vol 23 (1) ◽  
pp. 283-302 ◽  
Author(s):  
Matthias Bergner ◽  
Anna Dall’Acqua ◽  
Steffen Fröhlich
Keyword(s):  
Surfaces Of Revolution ◽  
Willmore Surfaces ◽  
Willmore Surfaces Of Revolution
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