The conformal Gauss map and the stability of Willmore surfaces

1991 ◽  
Vol 9 (3) ◽  
pp. 305-317 ◽  
Author(s):  
Bennett Palmer
Keyword(s):  
Gauss Map ◽  
Willmore Surfaces ◽  
The Stability ◽  
Conformal Gauss Map

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.

Open discussion

1982 ◽  
Vol 99 ◽  
pp. 605-613
Author(s):  
P. S. Conti
Keyword(s):  
Buenos Aires ◽  
Large Mass ◽  
List Type ◽  
Common Phenomenon ◽  
Open Discussion ◽  
The Stability ◽  
Ring Nebulae

Conti: One of the main conclusions of the Wolf-Rayet symposium in Buenos Aires was that Wolf-Rayet stars are evolutionary products of massive objects. Some questions:–Do hot helium-rich stars, that are not Wolf-Rayet stars, exist?–What about the stability of helium rich stars of large mass? We know a helium rich star of ∼40 MO. Has the stability something to do with the wind?–Ring nebulae and bubbles : this seems to be a much more common phenomenon than we thought of some years age.–What is the origin of the subtypes? This is important to find a possible matching of scenarios to subtypes.

Symmetric Multistep Methods Revisited: II. Numerical Experiments

1999 ◽  
Vol 173 ◽  
pp. 309-314 ◽  
Author(s):  
T. Fukushima
Keyword(s):  
Multistep Methods ◽  
Computational Time ◽  
Integration Time ◽  
Implicit Methods ◽  
Symmetric Methods ◽  
Celestial Bodies ◽  
The Stability ◽  
Predictor Corrector ◽  
Symmetric Multistep Methods ◽  
Integration Errors

AbstractBy using the stability condition and general formulas developed by Fukushima (1998 = Paper I) we discovered that, just as in the case of the explicit symmetric multistep methods (Quinlan and Tremaine, 1990), when integrating orbital motions of celestial bodies, the implicit symmetric multistep methods used in the predictor-corrector manner lead to integration errors in position which grow linearly with the integration time if the stepsizes adopted are sufficiently small and if the number of corrections is sufficiently large, say two or three. We confirmed also that the symmetric methods (explicit or implicit) would produce the stepsize-dependent instabilities/resonances, which was discovered by A. Toomre in 1991 and confirmed by G.D. Quinlan for some high order explicit methods. Although the implicit methods require twice or more computational time for the same stepsize than the explicit symmetric ones do, they seem to be preferable since they reduce these undesirable features significantly.

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