Wolf Prize Lecture - Rick Schoen (Stanford): The geometry of eigenvalue extremal problems

Thu, 08/06/201711:00-12:00
Levin building, lecture hall 8
Title: “The geometry of eigenvalue extremal problems”
Abstract: When we choose a metric on a manifold we determine the spectrum of
the Laplace operator. Thus an eigenvalue may be considered as a functional
on the space of metrics. For example the first eigenvalue would be the fundamental
vibrational frequency. In some cases the normalized eigenvalues are bounded
independent of the metric. In such cases it makes sense to attempt to find
critical points in the space of metrics. In this talk we will survey two cases in
which progress has been made focusing primarily on the case of surfaces with
boundary. We will describe the geometric structure of the critical metrics which
turn out to be the induced metrics on certain special classes of minimal (mean curvature
zero) surfaces in spheres and euclidean balls. The eigenvalue extremal problem is thus
related to other questions arising in the theory of minimal surfaces.