Variational Methods in Geometry Seminar

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. For surfaces, the critical metrics 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.In the first half of the talk we will give an overview of progress that has been made on the eigenvalue problem for surfaces with boundary, and contrast this with some recent results in higher dimensions. The second hour of the talk will focus on free boundary minimal surfaces the ball. References: A. Fraser, R. Schoen, Shape optimization for the Steklov problem in higher dimensions, arXiv:1711.043 A. Fraser, R. Schoen, Sharp eigenvalue bounds and minimal surfaces in the ball, Invent. Math. 203 (2016), no. 3, 823-890.