Minimal
            <i>n</i>
            -noids in hyperbolic and anti-de Sitter 3-space

Bobenko, Alexander I.; Heller, Sebastian; Schmitt, Nicolas

doi:10.1098/rspa.2019.0173

Cited by 3 publications

(1 citation statement)

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“…Note that in [16], [7], there are some different treatments of minimal surfaces in H 3 via loop group methods.…”

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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“…Note that in [16], [7], there are some different treatments of minimal surfaces in H 3 via loop group methods.…”

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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Willmore Surfaces

Pinkall,

Gross

2024

Compact Textbooks in Mathematics

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The analog for a surface $$f\colon M\to \mathbb {R}^3$$ f : M → ℝ 3 of the bending energy $$\int _a^b\kappa ^2\,ds$$ ∫ a b κ 2 d s is the Willmore functional$$W(f)=\int _M H^2\det $$ W ( f ) = ∫ M H 2 det .

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Higher solutions of Hitchin’s self-duality equations

Heller

2020

Journal of Integrable Systems

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Solutions of Hitchin’s self-duality equations correspond to special real sections of the Deligne–Hitchin moduli space—twistor lines. A question posed by Simpson in 1997 asks whether all real sections give rise to global solutions of the self-duality equations. An affirmative answer would in principle allow for complex analytic procedures to obtain all solutions of the self-duality equations. The purpose of this article is to construct counter examples given by certain (branched) Willmore surfaces in three-space (with monodromy) via the generalized Whitham flow. Though these sections do not give rise to global solutions of the self-duality equations on the whole Riemann surface M, they induce solutions on an open and dense subset of it. This suggest a connection between Willmore surfaces, i.e., rank 4 harmonic maps theory, with the rank 2 self-duality theory.

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Minimal n -noids in hyperbolic and anti-de Sitter 3-space

Cited by 3 publications

References 37 publications

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

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

Willmore Surfaces

Higher solutions of Hitchin’s self-duality equations

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