Hardt–Rosenberg minimal equatorial fillings problem

Hardt and Rosenberg asked in 1990 whether the Lawson Möbius band is the unique embedded non-orientable minimal surface in the three-sphere bounded by a great circle. Bernstein and Ketover’s preprint claims a negative answer.

September 2026 claimed resolution

Bernstein and Ketover claim that, for each $k\in\mathbb{N}$ and odd $m\ge 3$, there is an embedded non-orientable minimal surface with great-circle boundary, genus $mk+1$, and Euler number $2(k+1)$. Hence one real-analytic Jordan curve bounds infinitely many embedded minimal surfaces of distinct topological types, and every even genus $m\ge 4$ occurs with Euler number $4$. The result is an existence theorem, not a classification of all fillings; the claim is unverified.

annals.math.princeton.edu · eudml.org · numdam.org · people.disim.univaq.it · quantamagazine.org · page.math.tu-berlin.de · en.wikipedia.org · quantamagazine.org · arxiv.org · export.arxiv.org · ar5iv.labs.arxiv.org · arxiv.org · mathstodon.xyz · mathstodon.xyz · scientificamerican.com · mathstodon.xyz · quantamagazine.org

Current status (as of September 2026): Bernstein and Ketover claim the 1990 uniqueness question is false and provide infinitely many fillings, but the claim is unverified and classification of all equatorial fillings remains open.