Navigating the space of symmetric CMC surfaces

Heller, Lynn; Heller, Sebastian; Schmitt, Nicolas

doi:10.4310/jdg/1542423626

Cited by 22 publications

(40 citation statements)

References 31 publications

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“…Instead, they give rise to a generalized reflection symmetry of the family of λ-connections. In the special case when the section is i • ρ-real and τ -real or ρ-real, the corresponding (equivariant) harmonic map admits a reflection symmetry along a totally geodesic subspace of H 3 respectively S 3 , for a visualization in the latter case see [HHSc,Figures 1,3 & 4].…”

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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Section: )mentioning

confidence: 99%

Real Holomorphic Sections of the Deligne–Hitchin Twistor Space

Biswas

Heller

Röser³

2019

Commun. Math. Phys.

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“…This gives rise to a branched equivariant Willmore surfacef which is minimal in H 3 away from its intersection with the boundary at infinity [14,Section 5]. The counting of branch orders in [15,Theorem 3.3] also holds in the case of (equivariant) minimal surfacesf constructed by the τ -negative holomorphic sections s, as it only depends on the local analysis near the singular points, and branch orders are given by the vanishing order of the Higgs field. In particular, (for odd q), this yields that (with the notations of [15,Theorem 3.3 By Theorem 4.5 it remains to show that the Willmore energy off is bigger than 16π.…”

Section: The Lightcone Approach To Conformal Surface Geometrymentioning

confidence: 99%

“…The counting of branch orders in [15,Theorem 3.3] also holds in the case of (equivariant) minimal surfacesf constructed by the τ -negative holomorphic sections s, as it only depends on the local analysis near the singular points, and branch orders are given by the vanishing order of the Higgs field. In particular, (for odd q), this yields that (with the notations of [15,Theorem 3.3 By Theorem 4.5 it remains to show that the Willmore energy off is bigger than 16π. This can be seen as follows: The family of regular singular connections on the torus yields a equivariant Willmore surface f on the 4-punctured torus by the reconstruction method in [14,Section 5].…”

Section: The Lightcone Approach To Conformal Surface Geometrymentioning

confidence: 99%

Energy of sections of the Deligne–Hitchin twistor space

2020

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We study a natural functional on the space of holomorphic sections of the Deligne–Hitchin moduli space of a compact Riemann surface, generalizing the energy of equivariant harmonic maps corresponding to twistor lines. We show that the energy is the residue of the pull-back along the section of a natural meromorphic connection on the hyperholomorphic line bundle recently constructed by Hitchin. As a byproduct, we show the existence of a hyper-Kähler potentials for new components of real holomorphic sections of twistor spaces of hyper-Kähler manifolds with rotating $$S^1$$ S 1 -action. Additionally, we prove that for a certain class of real holomorphic sections of the Deligne–Hitchin moduli space, the energy functional is basically given by the Willmore energy of corresponding equivariant conformal map to the 3-sphere. As an application we use the functional to distinguish new components of real holomorphic sections of the Deligne–Hitchin moduli space from the space of twistor lines.

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“…Starting from Lawson's minimal surface ξ g,1 one can deform ξ g,1 by changing the value of the (constant) mean curvature to obtain an (experimentally constructed) 1-parameter family of embedded constant mean curvature surfaces (see Figure 4) having the symmetries of Lawson's ξ g,1 minimal surface. The Willmore energy profile [28] over this family is shown in Figure 5. The conformal types of these surfaces are "rectangular": the quotient P 1 of the surface under the cyclic g +1 fold symmetry has four branch points arranged in a rectangle.…”

Section: Higher Genus Outlookmentioning

confidence: 99%