Smooth long-time existence of Harmonic Ricci Flow on surfaces

We prove that at a finite singular time for the Harmonic Ricci Flow on a surface of positive genus both the energy density of the map component and the curvature of the domain manifold have to blow up simultaneously. As an immediate consequence, we obtain smooth long-time existence for the Harmonic Ricci Flow with large coupling constant.Comment: 29 pages, final version incorporating minor modifications based on comments from referees, to appear in J. London Math. So

Minimal surfaces - variational theory and applications

Minimal surfaces are among the most natural objects in Differential Geometry, and have been studied for the past 250 years ever since the pioneering work of Lagrange. The subject is characterized by a profound beauty, but perhaps even more remarkably, minimal surfaces (or minimal submanifolds) have encountered striking applications in other fields, like three-dimensional topology, mathematical physics, conformal geometry, among others. Even though it has been the subject of intense activity, many basic open problems still remain. In this lecture we will survey recent advances in this area and discuss some future directions. We will give special emphasis to the variational aspects of the theory as well as to the applications to other fields.Comment: Proceedings of the ICM, Seoul 201