Free boundary minimal surfaces in ellipsoids

In the Euclidean unit ball, the equatorial disc is unique in the class of immersed free boundary minimal discs [2]. In the '80s, Smyth [5, p. 411] raised the question whether or not free boundary minimal discs in Euclidean ellipsoids must also be planar. This question remained unresolved for nearly four decades until the discovery of several types of nonplanar free boundary minimal discs in ellipsoids [1,3,4]. Moreover, any ellipsoid contains at least three distinct free boundary minimal annuli [4].

References

• R. Haslhofer and D. Ketover, Free boundary minimal disks in convex balls. Preprint (arXiv:2307.01828).
• J. C. C. Nitsche, Stationary partitioning of convex bodies. Arch. Rational Mech. Anal. 89 (1985), 1–19.
• R. Petrides, Non planar free boundary minimal disks into ellipsoids. Preprint (arXiv:2304.12111).
• M. B. Schulz, Equivariant free boundary minimal discs and annuli in ellipsoids. Preprint (arXiv:2406.13465).
• B. Smyth, Stationary minimal surfaces with boundary on a simplex. Invent. Math. 76 (1984), 411–420.

More free boundary minimal discs

The ellipsoid with principal axes of length $3$, $2$, $1$ contains many distinct nonplanar free boundary minimal discs. The following animations indicate how these surfaces change when varying the length of the third principal axis.