Free boundary minimal surfaces and the Steklov eigenvalue problem (III)

If we fix a smooth compact surface $M$ with boundary, we can consider
all Riemannian metrics on $M$ with fixed boundary length. We can then
hope to find a canonical metric by maximizing a first eigenvalue. The
eigenvalue problem which turns out to lead to geometrically interesting
maximizing metrics is the Steklov eigenvalue problem; that is, the
Dirichlet-to-Neumann map on $\partial M$. There is a close connection
between this eigenvalue problem and minimal surfaces in a Euclidean ball
that are proper in the ball and meet the boundary of the ball
orthogonally. We refer to such minimal surfaces as free boundary
surfaces since they arise variationally as critical points of the area
among surfaces in the ball whose boundaries lie on $\partial B$ but are
free to vary on $\partial B$.

In this series of lectures I will introduce the eigenvalue problem, and
explain the connection with free boundary minimal surfaces. I will give
an overview of joint work with R. Schoen on existence and regularity of
metrics on a surface with boundary that maximize the first nonzero
normalized Steklov eigenvalue, and discuss existence, uniqueness and
compactness results for free boundary minimal surfaces in $B^n$.