Every nonflat conformal minimal surface is homotopic to a proper one

Author: Tjaša Vrhovnik
Location: Seminar Room 2 IMAG
Abstract: Given an open Riemann surface $M$ , we prove that every nonflat conformal minimal immersion $M\to {\mathbb{R}}^n$ ( $n\ge 3$ ) is homotopic through nonflat conformal minimal immersions $M\to {\mathbb{R}}^n$ to a proper one. If $n\ge 5$ , it may be chosen in addition injective, hence a proper conformal minimal embedding. Prescribing its flux, as a consequence, every nonflat conformal minimal immersion $M\to {\mathbb{R}}^n$ is homotopic to the real part of a proper holomorphic null embedding $M\to {\mathbb{C}}^n$ . We also obtain a result for a more general family of holomorphic immersions from an open Riemann surface into ${\mathbb{C}}^n$ directed by Oka cones in ${\mathbb{C}}^n$ .