A duality theorem for Willmore surfaces

Bryant, Robert L.

doi:10.4310/jdg/1214438991

Cited by 316 publications

(532 citation statements)

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“…2 (x, x)v ∞ which last is the expression for classical inverse-stereoprojection that can be found in [12].…”

Section: Symmetric R-spaces: Definition and Examplesmentioning

confidence: 93%

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Isothermic submanifolds of symmetric R-spaces

Burstall¹,

Donaldson²,

Pedit³

et al. 2011

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. We extend the classical theory of isothermic surfaces in conformal 3-space, due to Bour, Christoffel, Darboux, Bianchi and others, to the more general context of submanifolds of symmetric R-spaces with essentially no loss of integrable structure. IntroductionBackground. A surface in R 3 is isothermic if, away from umbilics, it admits coordinates which are simultaneously conformal and curvature line or, more invariantly, if it admits a holomorphic quadratic differential q which commutes with the (trace-free) second fundamental form (then the coordinates are z = x + iy for which q = dz 2 ). Examples include surfaces of revolution, cones and cylinders; quadrics and constant mean curvature surfaces (where q is the Hopf differential).Starting with the work of Bour [9, §54], isothermic surfaces were the focus of intensive study by the geometers of the late 19th and early 20th centuries with contributions from Christoffel, Cayley, Darboux, Demoulin, Bianchi, Calapso, Tzitzéica and many others. There has been a recent revival of interest in the topic thanks to Cieśliński-Goldstein-Sym [26] who pointed out the links with soliton theory: indeed, isothermic surfaces constitute an integrable system with a particularly beautiful and intricate transformation theory, some aspects of which we list below:Conformal invariance. Since the trace-free second fundamental form is invariant (up to scale) under conformal diffeomorphisms of R 3 , such diffeomorphisms preserve the class of isothermic surfaces. Thus isothermic surfaces are more properly to be viewed as surfaces in the conformal 3-sphere.Deformations. At least locally, isothermic surfaces admit a 1-parameter deformation preserving the conformal structure and trace-free second fundamental form: this is the T -transformation of Bianchi [3] and Calapso [19]. In fact, isothermic surfaces are characterised by the existence of such a deformation [23] (see [47,17,14] for modern treatments).Darboux transformations. According to Darboux [27], given an isothermic surface, one may locally construct a 4-parameter family of new isothermic surfaces. Analytically, this is accomplished by solving an integrable system of linear differential equations; geometrically, the two surfaces envelop a conformal Ribaucour sphere congruence 1 . These transformations, the Darboux transformations, are analogous to the Bäcklund transformations of constant curvature surfaces (indeed, specialise to the latter in certain circumstances [40,43] Curved flats. Curved flats were introduced by and are an integrable system defined on submanifolds of a symmetric space G/H which, in non-degenerate cases, coincides with the G/H-system of Terng [54]. In particular, the space S 3 × S 3 \ ∆ of pairs of distinct points in S 3 is a (pseudo-Riemannian) symmetric space for the diagonal action of the conformal diffeomorphism group and curved flats in this space are the same as pairs of isothermic surfaces related by a Darboux transformation [16].Discrete theory. Bobenko-Pinkall [6] show that the combinatorics of ...

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“…2 (x, x)v ∞ which last is the expression for classical inverse-stereoprojection that can be found in [12].…”

Section: Symmetric R-spaces: Definition and Examplesmentioning

confidence: 93%

“…There is a corresponding eigenbundle decomposition g = φ 11 Thus s Γ t (j, i) = Φs(j)Γ t+s (j, i)Φs(i) −1 , for all edges (j, i). 12 We retain our standing assumption that G is the adjoint group of g but much of the theory of curved flats carries through without it.…”

Section: Curved Flats and Their Transformationsmentioning

confidence: 99%

Isothermic submanifolds of symmetric R-spaces

Burstall¹,

Donaldson²,

Pedit³

et al. 2011

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…This is a very standard result which can be checked as follows: we know that φ ε is energy minimizing and hence that Sard's Theorem implies that the set V c := {s ∈ [0, M]/∃x ∈ Ω ε such that φ ε (x) = s and ∇φ ε (x) = 0} (critical values) is negligible (moreover it is also closed (1) -68 -…”

Section: Estimation Of Bmentioning

confidence: 99%

Removability of singularities of harmonic maps into pseudo-Riemannian manifolds

Hélein¹

2004

Annales De La Faculté Des Sciences De Toulouse Mathématiques

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“…Nayatani (private communication) has pointed out to us that an example written down by Kusner [44], Rosenberg-Toubiana [64,65], R. Bryant [6,7] with 4 flat ends, genus 0, and Gauss map of degree d = 3, gives strict inequality. On any surface the support function u = X · N satisfies Lu = 0, where L is defined in (7.2), and because the ends are flat, u is bounded.…”

Section: Theorem 74 ([54]) the Index Of Stability Ofmentioning

confidence: 99%