Lecture 13, a general survey of PDE problems related to geometry and physics

On January 23rd, we have discussed several examples of nonlinear PDE problems related to problems of geometry and physics. These were:

1. The Yamabe problem (given a compact Riemannian manifold $(M,g_0)$, is there a metric conformal to $g_0$ and having constant scalar curvature?). You can look up N. Trudinger’s paper Remarks concerning the conformal deformation of riemannian structures on compact manifolds in Ann. Scuola Norm. Sup. di Pisa (1968), and a newer one by S. Brendle and F.C. Margues, Recent progress on the Yamabe problem (available on arxiv.org).

2. Minimal surfaces and constant mean curvature surfaces. The existence of disk-type minimal surfaces, via a variational approach, is described in a precise and very readable fashion in Chapter 4 of vol. I of the book Minimal Surfaces by U. Dierkes, S. Hildebrandt, A. Kuester and O. Wohlrab. The solution of the Rellich conjecture on the existence of (at least) two different constant mean curvature surfaces having the same boundary contour is described e.g. in a