The Calabi–Yau problem for minimal surfaces with Cantor ends

A BSTRACT . We show that every connected compact or bordered Riemann surface contains a Cantor set whose complement admits a complete conformal minimal immersion in 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 C n with n ≥ 3 , for holomorphic Legendrian immersions into an arbitrary complex contact manifold, and for superminimal immersions into any self-dual or anti-self-dual Einstein four-manifold.