scholarly journals Minimal Bubbling for Willmore Surfaces

2020 ◽  
Author(s):  
Nicolas Marque
Keyword(s):  
Willmore Surface ◽  
Willmore Surfaces ◽  
Willmore Energy

Abstract In this paper, we build an explicit example of a minimal bubble on a Willmore surface, showing there cannot be compactness for Willmore immersions of Willmore energy above $16 \pi $. Additionally, we prove an inequality on the 2nd residue for limits of sequences of Willmore immersions with simple minimal bubbles. Doing so, we exclude some gluing configurations and prove compactness for immersed Willmore tori of energy below $12 \pi $.

scholarly journals The $$\mu $$-Darboux transformation of minimal surfaces

2019 ◽  
Vol 162 (3-4) ◽  
pp. 537-558
Author(s):  
K. Leschke ◽  
K. Moriya
Keyword(s):  
Minimal Surface ◽  
Darboux Transformation ◽  
Gauss Map ◽  
Willmore Surface ◽  
Willmore Surfaces ◽  
Isothermic Surfaces ◽  
Pointwise Limit ◽  
The Right ◽  
Associated Family ◽  
Darboux Transforms

Abstract The classical notion of the Darboux transformation of isothermic surfaces can be generalised to a transformation for conformal immersions. Since a minimal surface is Willmore, we can use the associated $$\mathbb { C}_*$$C∗-family of flat connections of the harmonic conformal Gauss map to construct such transforms, the so-called $$\mu $$μ-Darboux transforms. We show that a $$\mu $$μ-Darboux transform of a minimal surface is not minimal but a Willmore surface in 4-space. More precisely, we show that a $$\mu $$μ-Darboux transform of a minimal surface f is a twistor projection of a holomorphic curve in $$\mathbb { C}\mathbb { P}^3$$CP3 which is canonically associated to a minimal surface $$f_{p,q}$$fp,q in the right-associated family of f. Here we use an extension of the notion of the associated family $$f_{p,q}$$fp,q of a minimal surface to allow quaternionic parameters. We prove that the pointwise limit of Darboux transforms of f is the associated Willmore surface of f at $$\mu =1$$μ=1. Moreover, the family of Willmore surfaces $$\mu $$μ-Darboux transforms, $$\mu \in \mathbb { C}_*$$μ∈C∗, extends to a $$\mathbb { C}\mathbb { P}^1$$CP1 family of Willmore surfaces $$f^\mu : M \rightarrow S^4$$fμ:M→S4 where $$\mu \in \mathbb { C}\mathbb { P}^1$$μ∈CP1.

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.

scholarly journals WEIERSTRASS–KENMOTSU REPRESENTATION OF WILLMORE SURFACES IN SPHERES

2020 ◽  
pp. 1-25
Author(s):  
JOSEF F. DORFMEISTER ◽  
PENG WANG
Keyword(s):  
Riemann Surface ◽  
Mean Curvature ◽  
Harmonic Maps ◽  
Gauss Map ◽  
Exceptional Case ◽  
Willmore Surface ◽  
Willmore Surfaces ◽  
Gauss Maps ◽  
Conformal Gauss Map

A Willmore surface $y:M\rightarrow S^{n+2}$ has a natural harmonic oriented conformal Gauss map $Gr_{y}:M\rightarrow SO^{+}(1,n+3)/SO(1,3)\times SO(n)$ , which maps each point $p\in M$ to its oriented mean curvature 2-sphere at $p$ . An easy observation shows that all conformal Gauss maps of Willmore surfaces satisfy a restricted nilpotency condition, which will be called “strongly conformally harmonic.” The goal of this paper is to characterize those strongly conformally harmonic maps from a Riemann surface $M$ to $SO^{+}(1,n+3)/SO^{+}(1,3)\times SO(n)$ , which are the conformal Gauss maps of some Willmore surface in $S^{n+2}.$ It turns out that generically, the condition of being strongly conformally harmonic suffices to be associated with a Willmore surface. The exceptional case will also be discussed.

scholarly journals Stability of stationary points for one-dimensional Willmore energy with spatially heterogeneous term

2020 ◽  
pp. 132812
Author(s):  
Masaaki Uesaka ◽  
Ken-Ichi Nakamura ◽  
Keiichi Ueda ◽  
Masaharu Nagayama
Keyword(s):  
Stationary Points ◽  
One Dimensional ◽  
Willmore Energy

scholarly journals Isothermic constrained Willmore tori in 3-space

2021 ◽  
Author(s):  
Lynn Heller ◽  
Sebastian Heller ◽  
Cheikh Birahim Ndiaye
Keyword(s):  
Conformal Class ◽  
Willmore Energy

AbstractWe show that the homogeneous and the 2-lobe Delaunay tori in the 3-sphere provide the only isothermic constrained Willmore tori in 3-space with Willmore energy below $$8\pi $$ 8 π . In particular, every constrained Willmore torus with Willmore energy below $$8\pi $$ 8 π and non-rectangular conformal class is non-degenerated.

Explicit parametrizations of Willmore surfaces

2014 ◽  
Author(s):  
V. M. Vassilev ◽  
P. A. Djondjorov ◽  
E. Atanassov ◽  
M. Ts. Hadzhilazova ◽  
I. M. Mladenov
Keyword(s):  
Willmore Surfaces
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