Computation of free boundary minimal surfaces <i>via</i> extremal Steklov eigenvalue problems

Oudet, Èdouard; Kao, Chiu‐Yen; Osting, Braxton

doi:10.1051/cocv/2021033

Cited by 10 publications

(5 citation statements)

References 34 publications

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“…Item (iii) is again based on numerical simulations. In [48] Kao, Osting and Oudet developed numerical methods to maximize the scale-invariant first Steklov eigenvalue on surfaces of genus zero. Their results confirm that the corresponding free boundary minimal surfaces have prismatic (i. e. "bipyramidal") symmetry P b−2 for b ∈ {5, 7} boundary components respectively octahedral symmetry for b = 6 boundary components (cf.…”

Section: Heuristics and Motivationmentioning

confidence: 99%

“…Their results confirm that the corresponding free boundary minimal surfaces have prismatic (i. e. "bipyramidal") symmetry P b−2 for b ∈ {5, 7} boundary components respectively octahedral symmetry for b = 6 boundary components (cf. [48,Table 2]). While the convergence stated in (ii) forces the configuration of boundary components of Γ max b to become more and more homogeneous as b increases, it is important to note that Γ max b does not necessarily exhibit any symmetries.…”

Section: Heuristics and Motivationmentioning

confidence: 99%

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Infinitely many pairs of free boundary minimal surfaces with the same topology and symmetry group

Carlotto¹,

Schulz²,

Wiygul³

2022

Preprint

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The topology and symmetry group of a free boundary minimal surface in the three-dimensional Euclidean unit ball do not determine the surface uniquely. We provide pairs of non-isometric free boundary minimal surfaces having any sufficiently large genus g, three boundary components and antiprismatic symmetry group of order 4(g + 1). A. Parametrization of theKarcher-Scherk towers B. Analysis on asymptotically cylindrical surfaces C. Graphical deformations of immersed hypersurfaces D. Asymptotic behavior of the Ketover free boundary minimal surfaces 1

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Section: Heuristics and Motivationmentioning

confidence: 99%

Section: Heuristics and Motivationmentioning

confidence: 99%

Infinitely many pairs of free boundary minimal surfaces with the same topology and symmetry group

Carlotto¹,

Schulz²,

Wiygul³

2022

Preprint

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“…Using this connection, the author together with É. Oudet and C.Y. Kao computed many FBMS [OKO21]. Analogously, R. Petrides showed that the solution of an extremal Laplace-Beltrami eigenvalue problem generates a minimal isometric immersion into some d-sphere by first eigenfunctions [Pet14].…”

Section: Introductionmentioning

confidence: 96%

Extremal graph realizations and graph Laplacian eigenvalues

Osting¹

2022

Preprint

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For a regular polyhedron (or polygon) centered at the origin, the coordinates of the vertices are eigenvectors of the graph Laplacian for the skeleton of that polyhedron (or polygon) associated with the first (non-trivial) eigenvalue. In this paper, we generalize this relationship. For a given graph, we study the eigenvalue optimization problem of maximizing the first (non-trivial) eigenvalue of the graph Laplacian over non-negative edge weights. We show that the spectral realization of the graph using the eigenvectors corresponding to the solution of this problem, under certain assumptions, is a centered, unit-distance graph realization that has maximal total variance. This result gives a new method for generating unit-distance graph realizations and is based on convex duality. A drawback of this method is that the dimension of the realization is given by the multiplicity of the extremal eigenvalue, which is typically unknown prior to solving the eigenvalue optimization problem. Our results are illustrated with a number of examples.

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“…In particular, for surfaces of genus zero these immersions are embeddings and the target ball has dimension 3 (see [FS16]). As recent results for genus zero surfaces, in [GL20], the authors study more carefully the shape of these surfaces as the number of boundary components goes to +∞, while in [KOO21], the authors perform a numerical method for maximization of eigenvalues in order to make beautiful pictures of the maximal shapes.…”

Section: Introductionmentioning

confidence: 99%

Shape optimization for combinations of Steklov eigenvalues on Riemannian surfaces

Petrides

2024

Math. Z.

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We prove existence and regularity of metrics which minimize combinations of Steklov eigenvalues over metrics of unit perimeter on a surface with boundary. We show that there are free boundary minimal immersions into ellipsoids parametrized by eigenvalues, such that the coordinate functions are eigenfunctions with respect to the minimal metrics. This work generalizes Fraser-Schoen's and the author's maximization for one eigenvalue among metrics of unit perimeter on a surface giving free boundary minimal immersions into balls. We also generalize the previous results of critical metrics for one eigenvalue to any combination of eigenvalues from target balls to target ellipsoids.

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Computation of free boundary minimal surfaces via extremal Steklov eigenvalue problems

Cited by 10 publications

References 34 publications

Infinitely many pairs of free boundary minimal surfaces with the same topology and symmetry group

Infinitely many pairs of free boundary minimal surfaces with the same topology and symmetry group

Extremal graph realizations and graph Laplacian eigenvalues

Shape optimization for combinations of Steklov eigenvalues on Riemannian surfaces

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