Exploring the manifold of embeddings

• Applicant: Professor Dr. Philipp Reiter
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 567788778

The main objective of this proposal is to investigate geometrically, analytically, and numerically the manifold of embedded curves and (hyper) surfaces. In particular we aim at finding shortest paths between two given configurations within the same isotopy class.This problem is motivated by applications in the sciences, e.g., elasticity theory or computer graphics (keyframing). We build up on a Riemannian metric on the set of (sufficiently smooth) closed embedded curves which has been proposed and investigated recently; in fact, we have been able to establish both metric and geodesic completeness along with the existence of minimizing geodesics. The definition of this metric is inspired by the notion of tangent-point energies, a family of self-avoiding functionals which blow up if an embedded object degenerates. The first work packages relate to further investigating the aforementioned metric on curves. We would like to take first steps towards general statements on (sectional) curvature and the shape of geodesics. We also intend to consider a parametrization-invariant metric which gives rise to the shape space. Further work packages concern the definition of suitable metrics on surfaces. Here we propose two different approaches. The first one concerns a Finslerian setting which will require the surfaces to be less regular compared to the second one which focuses on a Riemannian metric which involves additional curvature-related higher-order terms.

• DFG Programme: Research Grants