Willmore surfaces in Riemannian manifolds

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Geometric variational problems arise naturally in various branches of mathematics, physics, biology and computer science. The most prominent examples are the isoperimetric problem and minimal surfaces. In this project we study higher order variational problems such as the Willmore functional and variants thereof. This specific functional arises for example in general relativity, biology and in the theory of image restoration. In the last few years the Willmore functional and its variants were extensively studied in Euclidean space and our goal here is to develop and partially extend the theory to arbitrary target manifolds since this is relevant in the above mentioned applications. More precisely, we want to study the effect of the ambient curvature on the geometric and analytical properties of the functionals. Our main motivation comes from applications to general relativity and we want to better understand the relation between the Willmore functional and the relevant physical quantities in this theory. In order to achieve our goals we have to extend the existing regularity, compactness and existence results for the functionals under consideration and since we are dealing with critical problems this requires a careful and delicate study of the underlying partial differential equations.