Date:

Thu, 08/06/201711:00-12:00

Location:

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.

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.