The Calabi-Yau problem for minimal surfaces with Cantor ends

Abstract

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.

Rev. Mat. Iberoam., to appear

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