Sequences of Willmore surfaces

Leschke, Katrin; Pedit, Franz

doi:10.1007/s00209-007-0214-4

Cited by 6 publications

(10 citation statements)

References 18 publications

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“…We explain a corresponding∂ and ∂ construction on the harmonic conformal Gauss map to obtain Willmore sequences, [LP08]. Using the Willmore sequence we get a unified proof of the classification results for Willmore spheres [Bry84, Mon00,Eji88], and Willmore tori with non-trivial normal bundle [LPP05].…”

Section: Willmore Sequencesmentioning

confidence: 99%

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Harmonic map methods for Willmore surfaces

Leschke¹

2011

Harmonic Maps and Differential Geometry

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Section: Willmore Sequencesmentioning

confidence: 99%

“…Theorem 10 ([LPP05], [LP08]). If f : S 2 → S 4 is a Willmore sphere, or if f : T 2 → S 4 is a Willmore torus with non-trivial normal bundle, then f is given by complex holomorphic data.…”

Section: Lemma 9 ([Lpp05]mentioning

confidence: 99%

Harmonic map methods for Willmore surfaces

Leschke¹

2011

Harmonic Maps and Differential Geometry

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“…For a Willmore surface in S 4 , the adjoint transform is essentially equivalent to the 2-step Bäcklund transform (which is just a suitable composition of two 1-step Bäcklund transforms) introduced in [5]. It has been showed in [23] that the mean curvature sphere congruences of 1-step Bäcklund transforms always extend smoothly across the possible branch points.…”

Section: For the Purpose Of Constructing Immersed Examples It Is Impmentioning

confidence: 99%

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

Ma¹,

Wang²,

Wang³

2017

J. Differential Geom.

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The classification of Willmore 2-spheres in the n-dimensional sphere S n is a long-standing problem, solved only when n = 3, 4 by Bryant, Ejiri, Musso and Montiel independently. In this paper we give a classification when n = 5. There are three types of such surfaces up to Möbius transformations: (1) super-conformal surfaces in S 4 ; (2) minimal surfaces in R 5 ; (3) adjoint transforms of super-conformal minimal surfaces in R 5 . In particular, Willmore surfaces in the third class are not S-Willmore (i.e., without a dual Willmore surface).To show the existence of Willmore 2-spheres in S 5 of type (3), we describe all adjoint transforms of a super-conformal minimal surfaces in R n and provide some explicit criterions on the immersion property. As an application, we obtain new immersed Willmore 2-spheres in S 5 and S 6 , which are not S-Willmore.

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“…Combining work of Simon [Sn] and of Bauer-Kuwert [BK] one has existence of closed Willmore surfaces of any prescribed genus. Recently quite some results have been obtained concerning Willmore minimisers in fixed conformal classes [KuS4,KuS5,LP,SchM]. Riviere [R] proved a far reaching regularity result.…”

Section: Clamped Plate Modelsmentioning

confidence: 99%

Nonlinear Questions in Clamped Plate Models

Grunau

2009

Milan J. Math.

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Abstract. The linear clamped plate boundary value problem is a classical model in mechanics. The underlying differential equation is elliptic and of fourth order. The latter is a peculiar feature with respect to which this equation differs from numerous equations in physics and engineering which are of second order. Concerning the clamped plate boundary value problem, "linear questions" may be considered as well understood. This changes completely as soon as one poses the simplest "nonlinear question": What can be said about positivity preserving? Does a plate bend upwards when being pushed upwards? It is known that the answer is "no" in general. However, there are many positivity issues as e.g. "almost positivity" to be discussed.Boundary value problems for the "Willmore equation" are nonlinear counterparts for the linear clamped plate equation. The corresponding energy functional involves curvature integrals over the unknown surface. The Willmore equation is of interest in mechanics, membrane physics and, in particular, in differential geometry. Quite far reaching results were achieved concerning closed surfaces. As for boundary value problems, by far less is known. These will be discussed in symmetric situations.

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Sequences of Willmore surfaces

Cited by 6 publications

References 18 publications

Harmonic map methods for Willmore surfaces

Harmonic map methods for Willmore surfaces

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

Nonlinear Questions in Clamped Plate Models

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