A properly embedded holomorphic disc in the ball with finite area and dense boundary

2017/09/30 by Franc Forstnerič, Franc Forstneric
Mathematics · #Analytic and geometric function theory #Ball (mathematics) #Boundary (topology) #Bounded function #Holomorphic and Operator Theory #Holomorphic function #Identity theorem #Meromorphic and Entire Functions #Unit sphere #Zero set #math.CV #math.DG #msc:32H02 #msc:37J55 #msc:53D10

paper · pdf · doi:10.1007/s00208-018-1686-8

published as Math. Ann., 373 (2019) 1-2, 719-742 · Math. Ann., to appear

openalex created_date 2017/09/15 · arxiv created 2018/05/09 · openalex publication_date 2018/05/12 · arxiv updated 2019/10/15 · openalex updated_date 2026/08/05

Abstract

In this paper we construct a properly embedded holomorphic disc in the unit ball \mathbbB² of ℂ² having a surprising combination of properties: on the one hand, it has finite area and hence is the zero set of a bounded holomorphic function on \mathbbB²; on the other hand, its boundary curve is everywhere dense in the sphere b\mathbbB².

Citations

• Bordered Riemann surfaces in C2
• Holomorphic Legendrian curves
• Differentiable Manifolds