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# Richard Schoen : An optimal eigenvalue problem and minimal surfaces in the ball

We consider the spectrum of the Dirichlet-Neumann map. This is the spectrum of the operator which sends a function on the boundary of a domain to the normal derivative of its harmonic extension. Along with the Dirichlet and Neumann spectrum, this problem has been much studied. We show how the problem of finding domains with fixed boundary area and largest first eigenvalue is connected to the study of minimal surfaces in the ball which meet the boundary orthogonally (free boundary solutions). We describe some conjectures on optimal surfaces and some progress toward their resolution. This is joint work with Ailana Fraser.

**Category**: Algebraic Geometry**Duration**: 01:34:32**Date**: May 9, 2011 at 3:55 PM**Views**: 102-
**Tags:**seminar, Algebraic Geometry Seminar

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