Geometric curvature energies

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The central issue of this research proposal is the study of geometric curvature energies on curves, surfaces, and higher-dimensional objects of low regularity. These energies exhibit self-avoidance and regularizing effects: a finite energy level guarantees and quantifies embeddedness of the geometric object, and leads to higher regularity. There is a variety of geometric curvature energies ranging from hard steric constraints to soft repulsive potentials, such as global curvature, integral versions of Menger curvature, or self-repulsive knot energies. We investigate the analytical properties of these energies and study their behaviour on families of curves or surfaces. The results will be applied to topologically constrained variational problems in the class of embeddings, and we examine the corresponding timedependent geometric flow problems.