Geometry of Willmore surfaces in Riemannian manifolds and applications to General Relativity

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Geometric variational problems and related partial differential equations are natural tools in order to study the geometry of Riemannian manifolds and to define physical quantities in the theory of general relativity. In this project we consider possible generalizations of the Willmore functional in Riemannian manifolds related to the Hawking mass in general relativity. We study surfaces that minimize this functional subject to certain constraints or surfaces satisfying related partial differential equations. The main focus is to investigate the influence of the geometry of the ambient manifold to the geometry of these surfaces. Often such relations have a physical interpretation when the ambient manifold is an initial data set for the Einstein equations. For example, the center of mass of an isolated gravitating system can be defined using surfaces minimizing area among all surfaces enclosing the same volume. These are the kind of effects we study for the Willmore functional. Previous work of the applicant suggests that there is a tight connection of the Willmore functional to the scalar curvature of the ambient space for small surfaces and to the mass and the center of mass of an isolated gravitating system in the large scale. We also consider versions of the Willmore functional which are expected to capture the linear momentum.