The Calabi-Yau problem for minimal surfaces with Cantor ends (Q6118199)

Summary: 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 \(n \geqslant 3\), for holomorphic Legendrian immersions into an arbitrary complex contact manifold, and for superminimal immersions into any selfdual or anti-self-dual Einstein four-manifold.