Bending-driven geometric variational problems: singularities, dynamics, and nonlocal effects

• Applicant: Dr. Fabian Rupp
• Subject Area: Mathematics
• Term: since 2026
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 580168218

Many shapes in nature arise as minimizers of energy functionals. Soap films, which minimize area, are classical examples; more advanced models, such as the Willmore functional, incorporate bending resistance. Such curvature-dependent bending energies appear across various fields, including materials science, image processing, general relativity, and cell biology. The mathematical analysis of these energies and their gradient flows is highly challenging, as they involve higher-order, scaling-critical partial differential equations that may lead to singularities or topological degeneracy. This proposal develops innovative techniques to understand such singular behavior, to describe the arising geometric structures, to rule out unphysical phenomena, and to extend solutions through singular regimes in a mathematically consistent manner. Our first research direction introduces a new notion of weak solutions for the Willmore flow, based on recent advancements in optimal transport. One focus is to continue solutions past singularities, allowing for substantially broader applications in geometric analysis and mathematical physics. Second, we study recently proposed fractional curvature energies, which capture long-range interactions. We establish a weak variational framework for studying the existence and regularity of minimizers and explore connections to classical local concepts. Third, we rigorously analyze variational models for biomembranes, refining them to avoid unphysical features such as self-interpenetration or branching and to enable a precise geometric description of singular processes like budding transitions during cell division. Finally, motivated by both theory and applications, we investigate corresponding free boundary problems that are connected to modeling incompressible organelles in biomechanics and to the Willmore conjecture in higher codimensions.

• DFG Programme: Emmy Noether Independent Research Groups