Computation of free boundary minimal surfaces via extremal Steklov eigenvalue problems
Computation of free boundary minimal surfaces via extremal Steklov eigenvalue problems
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DOI：
10.1051/cocv/2021033
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发表时间：
2020-07
期刊：
影响因子：
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通讯作者：
É. Oudet;C. Kao;B. Osting
中科院分区：
文献类型：
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作者：
É. Oudet;C. Kao;B. Osting
最近Fraser和Schoen证明了在一个有边界的紧致曲面上的某个极值Steklov特征值问题的解可以用来生成一个自由边界极小曲面，即，包含在球中的具有（i）零平均曲率和（ii）正交地与球的边界相交的表面（doi：10.1007/s 00222 -015-0604-x）。在本文中，我们开发的数值方法，使用这种连接实现自由边界极小曲面。也就是说，在一个具有亏格γ和B个边界分量的紧致曲面上，我们在一类光滑度量g上最大化σj（，g）L（，g），其中σj（，g）是第j个非零Steklov特征值，L（，g）是的长度。我们的数值方法包括（i）使用多连通域的共形一致化，以避免显式参数化的类度量，（ii）准确地解决边界加权Steklov特征值问题在多连通域，和（iii）开发基于梯度的优化方法，这个非光滑的特征值优化问题。对于亏格γ = 0和B = 2，.，9，12，15，20的边界分量，我们数值求解了第一特征值的极值Steklov问题。相应的特征函数生成一个自由边界极小曲面，我们显示在醒目的图像。对于更高的特征值，数值证据表明，最大化退化，但我们计算本地最大化的第二和第三特征值与B = 2边界组件和第三和第五特征值与B = 3边界组件。
Recently Fraser and Schoen showed that the solution of a certain extremal Steklov eigenvalue problem on a compact surface with boundary can be used to generate a free boundary minimal surface, i.e., a surface contained in the ball that has (i) zero mean curvature and (ii) meets the boundary of the ball orthogonally (doi:10.1007/s00222-015-0604-x). In this paper, we develop numerical methods that use this connection to realize free boundary minimal surfaces. Namely, on a compact surface, Σ, with genus γ and b boundary components, we maximize σj(Σ, g) L(∂Σ, g) over a class of smooth metrics, g, where σj(Σ, g) is the jth nonzero Steklov eigenvalue and L(∂Σ, g) is the length of ∂Σ. Our numerical method involves (i) using conformal uniformization of multiply connected domains to avoid explicit parameterization for the class of metrics, (ii) accurately solving a boundary-weighted Steklov eigenvalue problem in multi-connected domains, and (iii) developing gradient-based optimization methods for this non-smooth eigenvalue optimization problem. For genus γ = 0 and b = 2, …, 9, 12, 15, 20 boundary components, we numerically solve the extremal Steklov problem for the first eigenvalue. The corresponding eigenfunctions generate a free boundary minimal surface, which we display in striking images. For higher eigenvalues, numerical evidence suggests that the maximizers are degenerate, but we compute local maximizers for the second and third eigenvalues with b = 2 boundary components and for the third and fifth eigenvalues with b = 3 boundary components.