Free boundary minimal surfaces in toroids
Courant [1, II. § 3.2] wrote that “the most interesting problems with free boundaries are those in which the entire boundary is free on a given closed surface not of genus zero, e. g. on a torus.” More than 80 years later, infinitely many embedded, free boundary minimal annuli and Möbius bands have been discovered in strictly mean convex toroids [2]. The annuli have two sides (here coloured gold and blue) and make an even number of half-twists. In contrast, Möbius bands are nonorientable, having only one side (here coloured gold) with an odd number of half-twists. For comparison, nonorientable surfaces can not be properly embedded in the Euclidean unit ball for topological reasons.
References
- R. Courant, The existence of minimal surfaces of given topological structure under prescribed boundary conditions. Acta Math. 72 (1940), 51–98.
- M. B. Schulz, Free boundary minimal Möbius bands in toroids. Preprint (arXiv:2410.05781).
with symmetry axis (1 half-twist)
with symmetry axes (2 half-twists)
with symmetry axes (3 half-twists)
(4 half-twists)
(5 half-twists)
(8 half-twists)
(15 half-twists)
(22 half-twists)
Exotic examples
Free boundary minimal annuli and Möbius bands in toroids are highly nonunique even if one prescribes their full symmetry group. In fact we expect that there exist infinitely many embedded solutions with any given number of half-twists.
with symmetry axis (1 half-twist)
with symmetry axis (1 half-twist)
with symmetry axis (1 half-twist)
(3 half-twists)
(0 half-twists)
(0 half-twists)
with symmetry axes (2 half-twists)
