On the Björling problem for Willmore surfaces

Brander, David; Wang, Peng

doi:10.4310/jdg/1519959622

Cited by 5 publications

(16 citation statements)

References 28 publications

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“…But we also obtain several new examples of Willmore immersions from RP 2 to S 4 with various Willmore energies, which are different from the one of the Veronese surface. These surfaces are in fact minimal in S 4 and, to the authors' best knowledge, they are the first concrete known examples of minimal immersions from RP 2 to S 4 which are different from the Veronese surface.…”

Section: Introductionmentioning

confidence: 76%

“…As to (2) and (3), first we note that there exists no non-orientable translationally equivariant minimal surfaces in R 3 , since the Catenoid is the only translationally equivariant minimal surface in R 3 . Secondly, by Lemma 1.2 of [20] (see also [14]), y is conformally equivalent to a minimal surface in S 3 or H 3 if and only if so is its associated family.…”

Section: Equivariant Willmore Surfaces From Moebius Strips Into Spheresmentioning

confidence: 99%

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On symmetric Willmore surfaces in spheres II: The orientation reversing case

Dorfmeister

Wang

2020

Differential Geometry and its Applications

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In this paper we provide a systematic treatment of Willmore surfaces with orientation reversing symmetries and illustrate the theory by (old and new) examples. For finite order orientation reversing symmetries with fixed point as well as for finite order orientation reversing symmetries without fixed point the general theory is presented and concrete examples are given.Moreover, we apply our theory to isotropic Willmore two-spheres in S 4 and derive a necessary condition for such ( possibly branched) isotropic surfaces to descend to (possibly branched) maps from RP 2 to S 4 . The Veronese sphere and several other examples of non-branched Willmore immersions from RP 2 to S 4 are derived as an illustration of the general theory. The Willmore immersions of RP 2 , just mentioned and different from the Veronese sphere, are new to the authors' best knowledge. All these examples are conformally equivalent to minimal immersions into S 4 . We also provide a characterization of equivariant Willmore Moebius strips in S n+2 , and relate our results to Lawson's minimal Moebius strip in S 3 . Moreover, we present a defining characterization of all equivariant Willmore Klein bottles. Finally, we discuss orientation reversing symmetries of finite order which have a fixed point in the surface.

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Section: Introductionmentioning

confidence: 76%

Section: Equivariant Willmore Surfaces From Moebius Strips Into Spheresmentioning

confidence: 99%

On symmetric Willmore surfaces in spheres II: The orientation reversing case

Dorfmeister

Wang

2020

Differential Geometry and its Applications

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“…This problem has its roots in the classical Björling problem for minimal surfaces in R 3 , and has recently been examined for several combinations of Q 2+n and A 2 , generally with n = 1, see e.g. [4,6,13,3,1].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…These are precisely the ruled submanifolds without planar points and whose induced metric is flat, see Theorem 3. 4.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The geometric Cauchy problem for rank-one submanifolds

Raffaelli¹

2018

Preprint

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Given a smooth distribution D of m-dimensional planes along a smooth regular curve γ in R m+n , we consider the following problem: To find an m-dimensional developable submanifold of R m+n , that is, a ruled submanifold with constant tangent space along the rulings, such that its tangent bundle along γ coincides with D. In particular, we give sufficient conditions for the local well-posedness of the problem, together with a parametric description of the solution.

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“…Theorem 2.5. [53] (compare also [8,28,56]) Let y be a Willmore surface in S n+2 , with its normalized potential being of the form…”

Section: 22mentioning

confidence: 99%

Willmore deformations between minimal surfaces in $H^{n+2}$ and $S^{n+2}$

Wang,

Wang

2020

Preprint

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In this paper we show that locally there exists a Willmore deformation between minimal surfaces in S n+2 and minimal surfaces in H n+2 , i.e., there exists a smooth family of Willmore surfaces {yt, t ∈ [0, 1]} such that (yt)|t=0 is conformally equivalent to a minimal surface in S n+2 and (yt)|t=1 is conformally equivalent to a minimal surface in H n+2 . For some cases the deformations are global. Consider the Willmore deformations of the Veronese two-sphere and its generalizations in S 4 , for any positive number W0 ∈ R + , we construct complete minimal surfaces in H 4 with Willmore energy being equal to W0. An example of complete minimal Möbius strip in H 4 with Willmore energy 6 √ 5π 5≈ 10.733π is also presented. We also show that all isotropic minimal surfaces in S 4 admit Jacobi fields different from Killing fields, i.e., they are not "isolated".

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On the Björling problem for Willmore surfaces

Cited by 5 publications

References 28 publications

On symmetric Willmore surfaces in spheres II: The orientation reversing case

On symmetric Willmore surfaces in spheres II: The orientation reversing case

The geometric Cauchy problem for rank-one submanifolds

Willmore deformations between minimal surfaces in $H^{n+2}$ and $S^{n+2}$

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