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 $B^n$.

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