Analysis Seminar

I will discuss the connection between sharp eigenvalue bounds and minimal surfaces in two cases: The first eigenvalue of the Laplacian on a closed surface among unit area metrics, and the first Steklov eigenvalue on a compact surface with non empty boundary among metrics with unit length boundary. In both cases maximizing metrics - if they exist - are induced by certain minimal immersions. More precisely, minimal immersions into round spheres for the closed case and free boundary minimal immersions into Euclidean balls in the bordered case. I will discuss the solution of the existence problem for maximizers in both these cases, which provides many new examples of minimal surfaces of the aforementioned types. This is based on joint work with Anna Siffert in the closed case and Romain Petrides in the bordered case.