Romain Petrides

Université Paris Cité Université Paris Cité

Existence of non planar free boundary minimal disks into ellipsoids

Analysis and Geometry Seminar

23rd March 2023, 3:30 pm – 4:30 pm
Fry Building, Room 2.04

We show the existence of non planar embedded free boundary minimal disks into ellipsoids of R^3. A free boundary minimal disk D into a surface S of R^3 is a minimal surface with the topology of the disk such that D meets S orthogonally along the boundary. Of course, equatorial disks, which are planar satisfy this property on ellipsoids. This existence question was raised by Dierkes, Hildebrandt, Küster and Wohlrab in 1992.

It is a surprising result because by Nitsche in 1985, all free boundary minimal disks into ellipsoids are equatorial, and in an analogue problem: all simple closed geodesics in ellipsoids have to be equatorial. Our result is comparable to the recent answer of a question by Yau (1987) by Haslhofer and Ketover in 2019: there are non planar embedded minimal spheres into sufficiently elongated ellipsoids of R^4.

Our proof relies on a characterization of branched free boundary minimal immersions into ellipsoids as critical objects of functionals depending on Steklov eigenvalues with respect to a Riemannian metric on the surface. We obtain these non planar disks by maximization of well chosen linear combinations of the first and second Steklov eigenvalues among metrics on the disk with a fixed perimeter. We will also explain why we also build embedded disks by this method.

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