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

${\mathbb{C}}^n$

.