Abstract: The results of this talk on recent numerical aspects of metric optimization have been obtained in collaboration with Chiu-Yen Kao and Braxton Osting.
The first part of the talk is dedicated to Nash’s isometric embedding theorem for surfaces. We recall the impressive results obtained by HEVEA’s project for the flat torus (V. Borrelli, F. Lazarus, B. Thibert et al.) and illustrate how spectral formulation may lead to a new intrinsic approach which are not related to Gromov’s construction.
Following the theoretical results of Fraser and Schoen, we describe in a second part a numerical approach to approximate minimal surfaces in the ball that is surfaces (i) contained in the ball (ii) that have zero mean curvature and (ii) meet the boundary of the ball orthogonally. 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 have not been observed previously.
