Université libre de Bruxelles, 22nd January 2025
Isabel Fernández. Free Boundary Minimal Annuli in the Ball.
We will review some recent results in the theory of free boundary minimal surfaces in the unit ball in R3, and we will show the existence of a family of immersed, non rotational, free boundary minimal annuli. Their existence answers in the negative a problem of the theory that dates back to Nitsche in 1985, who claimed that such annuli could not exist. Joint work with Laurent Hauswirth and Pablo Mira.
Giada Franz. Free boundary minimal surfaces with low topological types in the unit ball.
A free boundary minimal surface (FBMS) in the three-dimensional Euclidean unit ball is a critical point of the area functional with respect to variations that constrain its boundary to the boundary of the ball (i.e., the unit sphere). Nitsche proved in 1985 that the equatorial disc is the only FBMS in the ball which is topologically a disc. It is then natural to ask what are the examples of FBMS with higher topology. In this talk, we will discuss recent existence results, which give a rather complete picture for low topological types (i.e. when the genus is less than one and the number of boundary components is less than two). Uniqueness results are still widely open.
Anna Skorobogatova. Regularity for semilinear variational problems with a topological constraint.
I will discuss the regularity of solutions to a class of semilinear free boundary problems in which admissible functions have a topological constraint, or spanning condition, on their 1-level set. This constraint forces the 1-level set, which is a free boundary, to behave like a surface with singularities, attached to a fixed boundary frame, in the spirit of the set-theoretic Plateau problem. Two such free boundary problems that have been well-studied are the minimization of capacity among surfaces sharing a common boundary and an Allen-Cahn approximation of the set-theoretic Plateau problem. We establish optimal Lipschitz regularity for solutions, and analytic regularity for the free boundaries away from a codimension two singular set. We further characterize the singularity models for these problems as conical critical points of the minimal capacity problem, which are closely related to spectral optimal partition and segregation problems. This is joint work with Mike Novack and Daniel Restrepo.