A Strong Parametric h-Principle for Complete Minimal Surfaces

We prove a parametric h-principle for complete nonflat conformal minimal immersions of an open Riemann surface M into R n , n ≥ 3 . It follows that the inclusion of the space of such immersions into the space of all nonflat conformal minimal immersions is a weak homotopy equivalence. When M is of fi...

Full description

Saved in:
Bibliographic Details
Published in:The Journal of geometric analysis 2025-02, Vol.35 (2), Article 42
Main Authors: Alarcón, Antonio, Lárusson, Finnur
Format: Article
Language:English
Subjects:
Abstract Harmonic Analysis
Convex and Discrete Geometry
Differential Geometry
Dynamical Systems and Ergodic Theory
Equivalence
Fourier Analysis
Global Analysis and Analysis on Manifolds
Mathematics
Mathematics and Statistics
Minimal surfaces
Riemann surfaces
Citations: Items that this one cites
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We prove a parametric h-principle for complete nonflat conformal minimal immersions of an open Riemann surface M into R n , n ≥ 3 . It follows that the inclusion of the space of such immersions into the space of all nonflat conformal minimal immersions is a weak homotopy equivalence. When M is of finite topological type, the inclusion is a genuine homotopy equivalence. By a parametric h-principle due to Forstnerič and Lárusson, the space of complete nonflat conformal minimal immersions therefore has the same homotopy type as the space of continuous maps from M to the punctured null quadric. Analogous results hold for holomorphic null curves M → C n and for full immersions in place of nonflat ones.
ISSN:1050-6926
1559-002X
DOI:10.1007/s12220-024-01873-6