A spectral curve approach to Lawson symmetric CMC surfaces of genus 2

Heller, Sebastian

doi:10.1007/s00208-014-1044-4

Cited by 27 publications

(61 citation statements)

References 19 publications

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“…which have the symmetry group of one of the Lawson surfaces; he also obtained constant mean curvature deformations of such surfaces in S 3 . See for example [He1], [He2] and [HeS]. The present work differs from these previous results in that now simply periodic solutions are considered, rather than doubly periodic solutions.…”

Section: Introductionmentioning

confidence: 86%

Spectral data for simply periodic solutions of the sinh-Gordon equation

Klein

2017

Differential Geometry and its Applications

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This note summarizes results that were obtained by the author in his habilitation thesis concerning the development of a spectral theory for simply periodic, 2-dimensional, complex-valued solutions u of the sinh-Gordon equation. Spectral data for such solutions are defined for periodic Cauchy data on a line (following Hitchin and Bobenko) and the space of spectral data is described by an asymptotic characterization. Using methods of asymptotic estimates, the inverse problem for the spectral data of such Cauchy data is answered. Finally a Jacobi variety for the spectral curve is constructed, and this is used to study the asymptotic behavior of the spectral data corresponding to actual simply periodic solutions of the sinh-Gordon equation on strips of positive height

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Section: Introductionmentioning

confidence: 86%

Spectral data for simply periodic solutions of the sinh-Gordon equation

Klein

2017

Differential Geometry and its Applications

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“…Note that after this choice of local lifts, the cocycle is uniquely determined as the section s is irreducible. Since C is a Stein manifold, the generalized Grauert theorem, [Bu], tells us that the cocycle is trivial, which enables us to find a global lift on C; see the proof of Theorem 8 in [He2] for more details.…”

Section: Define the Groupmentioning

confidence: 99%

“…Proof. This proposition follows from the fact that the SU(2) loop group Iwasawa decomposition is globally defined on the loop group [PrSe], and the section s is thus automatically admissible, see [He2,Theorem 6].…”

Section: )mentioning

confidence: 99%

Real Holomorphic Sections of the Deligne–Hitchin Twistor Space

Biswas

Heller

Röser³

2019

Commun. Math. Phys.

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We study the holomorphic sections of the Deligne-Hitchin moduli space of a compact Riemann surface, especially the sections that are invariant under the natural antiholomorphic involutions of the moduli space. Their relationships with the harmonic maps are established. As a by product, a question of Simpson on such sections, posed in [Si4], is answered.The other complex structure J on M SD arises from the identification of M SD with the moduli space of flat complex connections due to a result of Donaldson, [Do], in the case of rank n = 2 and by a result of Corlette in general [Co]. The third complex structure K is of course K = IJ = −JI. The significance of the self-duality equations in mathematics and theoretical physics is primarily due to, to quote Hitchin, the various incarnations of the solutions, bringing together geometry, topology, analysis and algebra in a harmonious way.Every hyper-Kähler manifold M has a twistor space associated to it. Topologically it is just the product of the space M with the 2-sphere:It has a tautological complex structure given by the natural complex structure on the 2sphere and the above complex structure xI p + yJ p + zK p on M × {(x, y, z)}; the natural projection of the twistor space to CP 1 is evidently holomorphic. The importance of the twistor space construction is due to the fact that the hyper-Kähler structure is encoded in holomorphic data on the twistor space. In the case of the moduli space of solutions to the self-duality equations, the corresponding twistor space, which is known as the Deligne-Hitchin moduli space and is denoted by M DH , admits another interpretation which is due to Deligne; see [Si2]. From Deligne's perspective, this Deligne-Hitchin moduli space M DH is considered as the moduli space of λ-connections on the Riemann surface Σ glued with the moduli space of λ-connections on the conjugate Riemann surface Σ via the Riemann-Hilbert isomorphism; for definitions and details see Section 1. As a consequence, integrable surfaces like harmonic maps and minimal surfaces can be described by sections of the twistor space M DH −→ CP 1 satisfying certain reality conditions that depend on what the target space is. We note that for the case of G C = SL(2, C), these surfaces are lying in any of the following: S 3 , H 3 and its quotients, and their Lorentzian counterparts.

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“…see [5,6]. In (2.2), Q is a holomorphic quadratic differential, the Hopf differential of the CMC surface, and ∇ S is the spin connection of the induced Levi-Civita connection, and ∇ S * is its dual connection.…”

Section: Monodromymentioning

confidence: 99%

“…Remark 4. Combining Theorem 3.2 with [6,Theorem 7] shows that deformations of compact CMC surfaces (with simple umbilics) that preserve the conjugacy classes of the monodromy representations of the associated families of flat connections must be induced by a family of dressing transformations. Dressing is a transformation of the family of flat connections induced by a λ-dependent gauge that becomes singular at certain spectral parameters λ 0 / ∈ S 1 ∪ {0, ∞} where parallel eigenlines with respect to ∇ λ0 exist.…”

Section: Deformations Of Cmc Surfacesmentioning

confidence: 99%

The asymptotic behavior of the monodromy representation of the associated family of a compact CMC surface

Heller

2016

Bull. London Math. Soc.

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Abstract. Constant mean curvature (CMC) surfaces in space forms can be described by their associated C * -family of flat SL(2, C)-connections ∇ λ . In this paper we consider the asymptotic behavior (for λ → 0) of the gauge equivalence classes of ∇ λ for compact CMC surfaces of genus g ≥ 2. We prove (under the assumption of simple umbilics) that the asymptotic behavior of the traces of the monodromy representation of ∇ λ determines the conformal type as well as the Hopf differential locally in the Teichmüller space.

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A spectral curve approach to Lawson symmetric CMC surfaces of genus 2

Cited by 27 publications

References 19 publications

Spectral data for simply periodic solutions of the sinh-Gordon equation

Spectral data for simply periodic solutions of the sinh-Gordon equation

Real Holomorphic Sections of the Deligne–Hitchin Twistor Space

The asymptotic behavior of the monodromy representation of the associated family of a compact CMC surface

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