Harmonic map methods for Willmore surfaces

@article{Leschke2010HarmonicMM,
  title={Harmonic map methods for Willmore surfaces},
  author={Katrin Leschke},
  journal={arXiv: Differential Geometry},
  year={2010}
}
  • K. Leschke
  • Published 17 March 2010
  • Physics
  • arXiv: Differential Geometry
In this note we demonstrate how the analogy between the harmonic Gauss map of a constant mean curvature surface and the harmonic conformal Gauss map of a Willmore surface can be used to obtain results on Willmore surfaces. 
Simple factor dressing and the López–Ros deformation of minimal surfaces in Euclidean 3-space
The aim of this paper is to investigate a new link between integrable systems and minimal surface theory. The dressing operation uses the associated family of flat connections of a harmonic map to
Dressing transformations of constrained Willmore surfaces
We use the dressing method to construct transformations of constrained Willmore surfaces in arbitrary codimension. An adaptation of the Terng--Uhlenbeck theory of dressing by simple factors to this
Darboux transforms and simple factor dressing of constant mean curvature surfaces
We define a transformation on harmonic maps $${N:\,M \to S^2}$$ from a Riemann surface M into the 2-sphere which depends on a parameter $${\mu \in \mathbb{C}_*}$$, the so-called μ-Darboux
N ov 2 01 3 DRESSING TRANSFORMATIONS OF CONSTRAINED WILLMORE
  • Mathematics
  • 2021
We use the dressing method to construct transformations of constrained Willmore surfaces in arbitrary codimension. An adaptation of the Terng–Uhlenbeck theory of dressing by simple factors to this
The $$\mu $$μ-Darboux transformation of minimal surfaces
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
Description of a mean curvature sphere of a surface by quaternionic holomorphic geometry (Submanifolds and Quaternion structure)
$Z:=\{C\in$ End$(\mathbb{H}^{2})|C^{2}=-$ Id $\}.$ This is the set of all quatenionic linear complex structures of $\mathbb{H}^{2}$ . Then two-spheres are parametrized by $\mathcal{Z}$ : Lemma 1

References

SHOWING 1-10 OF 26 REFERENCES
The tension field of the Gauss map
In this paper it is shown that the tension field of the Gauss map can be identified with the covariant derivative of the mean curvature vector field. Since a map with vanishing tension field is
The conformal Gauss map of submanifolds of the Möbius space
In this paper we study the conformal geometry of immersed submanifolds of the Möbius spaceSn introducing the conformal Gauss map. In particular we relate its harmonicity to an extended notion of
Sequences of Willmore surfaces
In this paper we develop the theory of Willmore sequences for Willmore surfaces in the 4-sphere. We show that under appropriate conditions this sequence has to terminate. In this case the Willmore
Conformal Geometry of Surfaces in S4 and Quaternions
Quaternions.- Linear algebra over the quaternions.- Projective spaces.- Vector bundles.- The mean curvature sphere.- Willmore Surfaces.- Metric and affine conformal geometry.- Twistor projections.-
Constrained Willmore surfaces
Constrained Willmore surfaces are conformal immersions of Riemann surfaces that are critical points of the Willmore energy $${\mathcal{W}} = \int H^2$$ under compactly supported infinitesimal
Willmore tori in the 4–Sphere with nontrivial normal bundle
We characterize Willmore tori in the 4-sphere with nontrivial normal bundle as Twistor projections of elliptic curves in complex projective space or as inverted minimal tori (with planar ends) in
Harmonic maps from a 2-torus to the 3-sphere
There have been many advances in recent years in the theory of harmonic maps of Riemann surfaces to spheres or symmetric spaces. These give constructions which produce harmonic maps from algebraic
Darboux transforms and spectral curves of constant mean curvature surfaces revisited
We study the geometric properties of Darboux transforms of constant mean curvature (CMC) surfaces and use these transforms to obtain an algebro-geometric representation of constant mean curvature
Quaternionic holomorphic geometry: Plücker formula, Dirac eigenvalue estimates and energy estimates of harmonic 2-tori
The paper develops the fundamentals of quaternionic holomorphic curve theory. The holomorphic functions in this theory are conformal maps from a Riemann surface into the 4-sphere, i.e., the
Conformal maps from a 2-torus to the 4-sphere
Abstract We study the space of conformal immersions of a 2-torus into the 4-sphere. The moduli space of generalized Darboux transforms of such an immersed torus has the structure of a Riemann
...
...