The Calabi-Yau problem for minimal surfaces with Cantor ends

We show that every connected compact or bordered Riemann surface contains a Cantor set whose complement admits a complete conformal minimal immersion in $\mathbb R^3$ with bounded image. The analogous result holds for holomorphic immersions into any complex manifold of dimension at least $2$, for holomorphic null immersions into $\mathbb C^n$ with $n\ge 3$, for holomorphic Legendrian immersions into an arbitrary complex contact manifold, and for superminimal immersions in any self-dual or anti-self-dual Einstein four-manifold.