For every open Riemann surface X, does there exist a complete conformal minimal immersion F:X→B^3⊂R^3 that is proper, where B^3 is the unit ball and completeness refers to the metric induced by F?

Calabi–Yau conjecture for complete bounded minimal surfaces

For every open Riemann surface $X$ , does there exist a complete conformal minimal immersion $F:X\to\mathbb{B}^{3}\subset\mathbb{R}^{3}$ that is proper, where $\mathbb{B}^{3}$ is the unit ball and completeness refers to the metric induced by $F$ ?

The problem asks which complete bounded minimal surfaces can exist, including the unresolved general immersed case. Known results settle important embedded and finite-topology settings, but not the full conjecture.

Known results

Colding and Minicozzi proved that every complete embedded minimal disk in $\mathbb{R}^{3}$ is proper; this yields the embedded-disk Calabi conjectures and finite-topology consequences.
Meeks, Pérez, and Ros proved the finite-genus case when the set of ends is countable.
Complete bounded minimal immersions realize broad classes of open Riemann-surface structures, including complements of countably many discs and Cantor sets.

October 2026 symmetric constructions

Antonio Alarcón and Franc Forstnerič report proper equivariant conformal minimal immersions under an explicit period condition and construct complete bounded examples with finite-group symmetry. The result applies to bordered Riemann surfaces with free boundary action and does not settle the general Calabi–Yau or Sullivan problems; independent corroboration was not found.

Current status (as of October 2026): embedded finite-topology cases and several broad immersed constructions are known, while the general complete bounded minimal-surface problem remains open; the symmetric advance is claimed progress and unverified.

Calabi–Yau conjecture for complete bounded minimal surfaces

Progress summary

Solutions 0