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 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.

scholarly journals Isothermic surfaces obtained from harmonic maps in S6

2018 ◽  
Vol 103 (117) ◽  
pp. 175-180
Author(s):  
Rui Pacheco
Keyword(s):  
Riemann Surface ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Harmonic Maps ◽  
Dimensional Sphere ◽  
Constant Mean Curvature Surfaces ◽  
Isothermic Surfaces ◽  
Smooth Map

The harmonicity of a smooth map from a Riemann surface into the 6-dimensional sphere S6 amounts to the closeness of a certain 1-form that can be written in terms of the nearly K?hler structure of S6. We will prove that the immersions F in R7 obtained from superconformal harmonic maps in S3 ? S6 by integration of the corresponding closed 1-forms are isothermic. The isothermic surfaces so obtained include a certain class of constant mean curvature surfaces in R3 that can be deformed isometrically through isothermic surfaces into non-spherical pseudo-umbilical surfaces in R7.

EXTRINSIC GEOMETRY OF WORLD-SHEET SUPERSYMMETRY THROUGH GENERALIZED SUPER-GAUSS MAPS

1992 ◽  
Vol 07 (24) ◽  
pp. 5995-6011 ◽  
Author(s):  
K.S. VISWANATHAN ◽  
R. PARTHASARATHY
Keyword(s):  
Riemann Surface ◽  
Extrinsic Curvature ◽  
Gauss Map ◽  
World Sheet ◽  
Invariant Action ◽  
Extrinsic Geometry ◽  
The World ◽  
Integrability Conditions ◽  
Gauss Maps

The extrinsic geometry of N=1 world-sheet supersymmetry is studied through generalized super-Gauss map. The world sheet, realized as a conformally immersed super-Riemann surface S in Rn (n=3 is studied for simplicity) is mapped into the supersymmetric Grassmannian G2,3. In order for the Grassmannian fields to form (super) tangent planes to S, certain integrability conditions are satisfied by G2,n fields. These conditions are explicitly derived. The supersymmetric invariant action for the Kähler σ-model G2,3 is reexpressed in terms of the world-sheet coordinates, thereby an off-shell supersymmetric generalization of the action proportional to the extrinsic curvature of the immersed surface is obtained.

scholarly journals Gauss map and the topology of constant mean curvature hypersurfaces of $$\mathbb {S}^{7}$$ and $$\mathbb {CP}^{3}$$

2019 ◽  
Vol 163 (1-2) ◽  
pp. 279-290
Author(s):  
Fidelis Bittencourt ◽  
Pedro Fusieger ◽  
Eduardo R. Longa ◽  
Jaime Ripoll
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Gauss Map ◽  
Constant Mean Curvature Hypersurfaces
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