Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves. Issue 4 (9th September 2015)

• Record Type: Journal Article
• Authors: Alarcón, A.; Drinovec Drnovšek, B.; Forstnerič, F.; López, F. J.
• Is Part Of: Proceedings of the London Mathematical Society. Volume 111:Issue 4(2015:Oct.)
• Page Start: 851
• Page End: 886
• Publication Date: 2015-09-09
• Subjects: Mathematics -- Periodicals; Mathematics; Periodicals; 510
• DOI: 10.1112/plms/pdv044
• Languages: English
• ISSNs: 0024-6115

Abstract

Abstract : In this paper, we find approximate solutions of certain Riemann–Hilbert boundary value problems for minimal surfaces inR n and null holomorphic curves inC n for anyn ⩾ 3 . With this tool in hand, we construct complete conformally immersed minimal surfaces inR n which are normalized by any given bordered Riemann surface and have Jordan boundaries. We also furnish complete conformal proper minimal immersions from any given bordered Riemann surface to any smoothly bounded, strictly convex domain ofR n which extend continuously up to the boundary; forn ⩾ 5, we find embeddings with these properties.