Every conformal minimal surface in ℝ³ is isotopic to the real part of a holomorphic\break null curve

Abstract

We show that for every conformal minimal immersion u : M → R 3 from an open Riemann surface M to R 3 there exists a smooth isotopy u t : M → R 3 (t ∈ [0, 1]) of conformal minimal immersions, with u 0 = u, such that u 1 is the real part of a holomorphic null curve M → C 3 (i.e. u 1 has vanishing flux). If furthermore u is nonflat then u 1 can be chosen to have any prescribed flux and to be complete. Keywords Riemann surfaces, minimal surfaces, holomorphic null curves. MSC (2010):53C42; 32B15, 32H02, 53A10. The main resultsLet M be a smooth oriented surface. A smooth immersion u = (u 1 , u 2 , u 3 ) : M → R 3 is minimal if its mean curvature vanishes at every point. The requirement that an immersion u be conformal uniquely determines a complex structure on M . Finally, a conformal immersion is minimal if and only if it is harmonic:of an open Riemann surface to C 3 is said to be a null curve if its differential dF = (dF 1 , dF 2 , dF 3 ) satisfies the equationThe real and the imaginary part of a null curve M → C 3 are conformal minimal immersions M → R 3 . Conversely, the restriction of a conformal minimal immersion u : M → R 3 to any simply connected domain Ω ⊂ M is the real part of a holomorphic null curve Ω → C 3 ; u is globally the real part of a null curve if and only if its conjugate differential d c u satisfiesThis period vanishing condition means that u admits a harmonic conjugate v, and F = u + ıv : M → C 3 (ı = √ −1) is then a null curve.In this paper we prove the following result which further illuminates the connection between conformal minimal surfaces in R 3 and holomorphic null curves in C 3 . We shall systematically use the term isotopy instead of the more standard regular homotopy when talking of smooth 1-parameter families of immersions. The analogous result holds for minimal surfaces in R n for any n ≥ 3, and the tools used in the proof are available in that setting as well. On a compact bordered Riemann surface we also have an up to the boundary version of the same result (cf. Theorem 4.1).

New Complex Analytic Methods in the Theory of Minimal Surfaces: A Survey

“…(Here M 1 ⋐ M 2 ⋐ · · · is an exhaustion of M as in (4.1).) To ensure completeness of X = lim i→∞ X i = ( X i,1 , X i,2 , X 3 ) : M → R 3 we suitably enlarge the intrinsic diameter of each immersion X i : M i → R 3 by using a Jorge-Xavier type labyrinth of compact sets inM i \ M i−1 as in the proof of Theorem 4.5. ized Gauss map of X 0 is nondegenerate [13], and (c) a complete conformal minimal immersion with vanishing flux such that all maps X t have the same generalized Gauss map M → CP n−1 (see [19,Corollary 1.4]).…”

“…(a) a complete conformal minimal immersion[13], (b) a complete conformal minimal immersion with arbitrary flux if the general-…”

Representing de Rham cohomology classes on an open Riemann surface by holomorphic forms

“…, and a sequence of numbers ǫ j > 0, j ≥ 1, such that the following properties are satisfied for all j. 1], then h| D j takes its values in Y and is nondegenerate.…”

“…Indeed, assume for a moment that such sequences exist. By (1), (2 j ), and (5 j ), there is a limit homotopy of holomorphic maps f t p = lim j→∞ f t p,j : X → A, (p, t) ∈ P × [0, 1], such that f t p −f t p,j D j < 2ǫ j for all (p, t) ∈ P ×[0, 1] and all j ≥ 1. Thus, properties (6 j ) ensure that f t p ∈ O * (X, Y ) for all (p, t) ∈ P × [0, 1]; take into account (1).…”

“…, ζ N ) is a parameter in B on which the maps h t p,ζ depend holomorphically. In the proof of [7, Theorem 4.1], the period-domination follows from [2, Lemma 5.1] (see also [1,Lemma 3.6]) with respect to a fixed basis of the first homology group H 1 (D j−1 , Z). In our case, we use exactly the same argument but apply the following generalisation of [2, Lemma 5.1].…”