Constrained Willmore surfaces of normalized mean curvature: The large genus blowup

• Applicant: Christian Scharrer, Ph.D.
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 569615323

Motivated by the bubbling phenomena of biological cell membranes, we investigate Willmore minimizers in the class of closed surfaces smoothly embedded in Euclidean 3-space with prescribed genus, total scalar mean curvature, and area. Since the Willmore functional is invariant under scaling, the last two constraints may be combined to a single constraint on their scaling invariant ratio. This natural constraint on the normalized mean curvature, which leads to a variety of previously unknown geometrically intriguing shapes, ranges between two characteristic bubble formations: Vanishing normalized mean curvature caused by the cancellation of two oppositely oriented concentric spheres, and the normalized mean curvature of two spheres touching in a single point. Given a constraint for the normalized mean curvature in this range, we obtain a sequence of smooth constrained Willmore minimizers as a function of their genus. We conjecture that after rescaling the sequence to have constant area, the varifold limit is given by two intersecting round spheres where the intersection angle depends smoothly on the constraint. More precisely, the prescribed topology concentrates in terms of small catenoidal tunnels along the circle of intersection. Instead of normalizing the area, one may alternatively translate and rescale the sequence to have unit supremum norm of the second fundamental form attained at the origin. This leads to a complete non-compact surface in the limit. We conjecture that this large genus blowup is a singly periodic Scherk surface asymptotic to two planes which intersect at the same angle as the two spheres, meaning that the whole one parameter family of Scherk’s singly periodic minimal surfaces can be obtained as large genus blowups of constrained Willmore surfaces. In this project, we propose to investigate this phenomenal conjecture as well as to study naturally arising related problems.

• DFG Programme: Emmy Noether Independent Junior Research Groups