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The parametric h-principle for minimal surfaces in R^n and null curves in C^n

Banff International Research Station for Mathematical Innovation and Discovery

Let $M$ be an open Riemann surface. It was proved by Alarc{\'o}n and Forstneri{\v c} that every conformal minimal immersion $M\to\mathbb R^3$ is isotopic to the real part of a holomorphic null curve $M\to\mathbb C^3$. We prove the following substantially stronger result in this direction: for any $n\ge 3$, the inclusion of the space of real parts of nonflat null holomorphic immersions $M\to\mathbb C^n$ into the space of nonflat conformal minimal immersions $M\to \mathbb R^n$ satisfies the parametric h-principle with approximation; in particular, it is a weak homotopy equivalence. Analogous results hold for several other related maps. For an open Riemann surface $M$ of finite topological type, we obtain optimal results by showing that the above inclusion and several related maps are inclusions of strong deformation retracts; in particular, they are homotopy equivalences. (Joint work with Finnur L{\'a}russon.)

Mathematics; Several complex variables and analytic spaces; Potential theory; Global analysis

video/mp4

Author affiliation: University of Ljubljana

2017-02-08

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/59619

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/