Proper superminimal surfaces of given conformal types in the hyperbolic four-space

Abstract

Let $H^4$ denote the hyperbolic four-space. Given a bordered Riemann surface, $M$, we prove that every smooth conformal superminimal immersion $\overline{M}\to H^4$ can be approximated uniformly on compacts in $M$ by proper conformal superminimal immersions $M\to H^4$. In particular, $H^4$ contains properly immersed conformal superminimal surfaces normalised by any given open Riemann surface of finite topological type without punctures. The proof uses the analysis of holomorphic Legendrian curves in the twistor space of $H^4$.

References in corpus (11)

• Null curves and directed immersions of open Riemann surfaces
• Every bordered Riemann surface is a complete proper curve in a ball
• The Poletsky-Rosay theorem on singular complex spaces
• New complex analytic methods in the theory of minimal surfaces: a survey
• Holomorphic Legendrian curves
• Darboux charts around holomorphic Legendrian curves and applications
• The Calabi-Yau property of superminimal surfaces in self-dual Einstein four-manifolds
• The Calabi-Yau problem for Riemann surfaces with finite genus and countably many ends
• Mergelyan approximation theorem for holomorphic Legendrian curves
• Holomorphic Legendrian curves in ${\mathbb{CP}}^3$ and superminimal surfaces in ${\mathbb{S}}^4$
• Minimal surfaces in minimally convex domains

Cited by in corpus (1)

• Recent developments on Oka manifolds