Classification of Willmore two-spheres in the 5-dimensional sphere

Ma, Xiang; Wang, Changping; Wang, Peng

doi:10.4310/jdg/1497405626

Cited by 6 publications

(5 citation statements)

References 30 publications

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“…After the first version of this work, we saw that there was a version under press of a generalisation to S 5 of Bryant's classification ( [21]). As there are also papers under review in the cases S n (with n ≥ 6), and for obvious size limitation, we will not discuss these cases.…”

Section: Remarkmentioning

confidence: 99%

The Classification of Branched Willmore Spheres in the $3$-Sphere and the $4$-Sphere

Michelat¹,

Rivière²

2017

Preprint

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We extend the classification of Robert Bryant of Willmore spheres in S 3 to variational branched Willmore spheres S 3 and show that they are inverse stereographic projections of complete minimal surfaces with finite total curvature in R 3 and vanishing flux. We also obtain a classification of variational branched Willmore spheres in S 4 , generalising a theorem of Sebástian Montiel. As a result of our asymptotic analysis at branch points, we obtain an improved C 1,1 regularity of the unit normal of variational branched Willmore surfaces in arbitrary codimension. We also prove that the width of Willmore sphere min-max procedures in dimension 3 and 4, such as the sphere eversion, is an integer multiple of 4π.

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Section: Remarkmentioning

confidence: 99%

The Classification of Branched Willmore Spheres in the $3$-Sphere and the $4$-Sphere

Michelat¹,

Rivière²

2017

Preprint

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“…We should mention that there is, as of yet, no classification of Willmore 2‐spheres in arbitrary codimension, even though partial results can be found in . These results indicate that the general classification of Willmore 2‐spheres in higher codimension is more involved and, even if resolved, selecting the homogeneous ones from such a list may be unpractical.…”

Section: Introductionmentioning

confidence: 92%

Möbius homogeneous Willmore 2-spheres

Pedit

Wang

2018

Bull. London Math. Soc.

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“…Superconformal surfaces. In the conformal n-sphere S n there is a class of surfaces called superconformal surfaces that arise in the study of Willmore surfaces (see [8,13,14,15,25]). These are the surfaces f for which II 1,0 V is isotropic, where V is the central sphere congruence of f .…”

Section: 2mentioning

confidence: 99%

Quartic differentials and harmonic maps in conformal surface geometry

Burstall¹,

Musso²,

Pember³

2020

Preprint

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We consider codimension 2 sphere congruences in pseudo-conformal geometry that are harmonic with respect to the conformal structure of an orthogonal surface. We characterise the orthogonal surfaces of such congruences as either S-Willmore surfaces, quasi-umbilical surfaces, constant mean curvature surfaces in 3-dimensional space forms or surfaces of constant lightlike mean curvature in 3-dimensional lightcones. We then investigate Bryant's quartic differential in this context and show that generically this is divergence free if and only if the surface under consideration is either superconformal or orthogonal to a harmonic congruence of codimension 2 spheres. We may then apply the previous result to characterise surfaces with such a property.

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Classification of Willmore two-spheres in the 5-dimensional sphere

Cited by 6 publications

References 30 publications

The Classification of Branched Willmore Spheres in the $3$-Sphere and the $4$-Sphere

The Classification of Branched Willmore Spheres in the $3$-Sphere and the $4$-Sphere

Möbius homogeneous Willmore 2-spheres

Quartic differentials and harmonic maps in conformal surface geometry

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