Complete Non-Orientable Minimal Surfaces in ℝ<sup>3</sup> and Asymptotic Behavior

Alarcón, Antonio; López, Francisco J.

doi:10.2478/agms-2014-0007

Cited by 3 publications

(1 citation statement)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By a different technique, relying on Runge-Mergelyan type theorems (cf. [8,12]), Alarcón and López obtained analogous results for nonorientable minimal surfaces in R 3 [13], null holomorphic curves in C 3 , and complex curves in C 2 [10]. (Recall that a null curve in C n , n ≥ 3, is a complex curve whose real and imaginary parts are minimal surfaces in R n .)…”

Section: Introductionmentioning

confidence: 74%

Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves: Figure. 5.1.

Alarcón

Drnovsek

Forstnerič

et al. 2015

Proc. London Math. Soc.

View full text Add to dashboard Cite

In this paper we find approximate solutions of certain Riemann-Hilbert boundary value problems for minimal surfaces in R n and null holomorphic curves in C n for any n ≥ 3. With this tool in hand we construct complete conformally immersed minimal surfaces in R n which are normalized by any given bordered Riemann surface and have Jordan boundaries. We also furnish complete conformal proper minimal immersions from any given bordered Riemann surface to any smoothly bounded, strictly convex domain of R n which extend continuously up to the boundary; for n ≥ 5 we find embeddings with these properties.

show abstract

Section: Introductionmentioning

confidence: 74%

Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves: Figure. 5.1.

Alarcón

Drnovsek

Forstnerič

et al. 2015

Proc. London Math. Soc.

View full text Add to dashboard Cite

show abstract

New Complex Analytic Methods in the Study of Non-Orientable Minimal Surfaces in ℝⁿ

2020

Self Cite

View full text Add to dashboard Cite

42 4.2. A Mergelyan theorem with fixed components 48 Chapter 5. A general position theorem for non-orientable minimal surfaces 55 Chapter 6. Applications 59 6.1. Proper non-orientable minimal surfaces in R n 59 6.2. Complete non-orientable minimal surfaces with fixed components 63 6.3. Complete non-orientable minimal surfaces with Jordan boundaries 68 6.4. Proper non-orientable minimal surfaces in p-convex domains 69 Bibliography 73iii AbstractThe aim of this work is to adapt the complex analytic methods originating in modern Oka theory to the study of non-orientable conformal minimal surfaces in R n for any n ≥ 3. These methods, which we develop essentially from the first principles, enable us to prove that the space of conformal minimal immersions of a given bordered non-orientable surface to R n is a real analytic Banach manifold (see Theorem 1.1), obtain approximation results of Runge-Mergelyan type for conformal minimal immersions from non-orientable surfaces (see Theorem 1.2 and Corollary 1.3), and show general position theorems for non-orientable conformal minimal surfaces in R n (see Theorem 1.4). We also give the first known example of a properly embedded non-orientable minimal surface in R 4 ; a Möbius strip (see Example 6.1).All our new tools mentioned above apply to non-orientable minimal surfaces endowed with a fixed choice of a conformal structure. This enables us to obtain significant new applications to the global theory of non-orientable minimal surfaces. In particular, we construct proper non-orientable conformal minimal surfaces in R n with any given conformal structure (see Theorem 1.6 (i)), complete non-orientable minimal surfaces in R n with arbitrary conformal type whose generalized Gauss map is nondegenerate and omits n hyperplanes of CP n−1 in general position (see Theorem 1.6 (iii)), complete non-orientable minimal surfaces bounded by Jordan curves (see Theorem 1.5), and complete proper non-orientable minimal surfaces normalized by bordered surfaces in p-convex domains of R n (see Theorem 1.7).

show abstract

Approximation theory for nonorientable minimal surfaces and applications

Alarcón

López

2015

Geom. Topol.

Self Cite

View full text Add to dashboard Cite

We prove a version of the classical Runge and Mergelyan uniform approximation theorems for non-orientable minimal surfaces in Euclidean 3-space R 3 . Then, we obtain some geometric applications. Among them, we emphasize the following ones:• A Gunning-Narasimhan type theorem for non-orientable conformal surfaces.• An existence theorem for non-orientable minimal surfaces in R 3 , with arbitrary conformal structure, properly projecting into a plane. • An existence result for non-orientable minimal surfaces in R 3 with arbitrary conformal structure and Gauss map omitting one projective direction.Finally, in a different line of applications, we prove an extension of the classical Gunning-Narasimhan theorem [18] (see also [21]); more specifically, we show that, for any open Riemann surface N and any antiholomorphic involution I : N → N without fixed points, there exist holomorphic 1-forms ϑ on N with I * ϑ = ϑ and prescribed periods and canonical divisor (see Theorem 6.4).Outline of the paper. The necessary notation and background on non-orientable minimal surfaces in R 3 is introduced in Sec. 2. In Sec. 3 we describe the compact subsets involved in the Mergelyan type approximation, and define the notion of conformal non-orientable minimal immersion from such a subset into R 3 . In Sec. 4 we prove several preliminary approximation results that flatten the way to the proof of the main theorem in Sec. 5. Finally, the applications are derived in Sec. 6. PreliminariesLet · denote the Euclidean norm in K n (K = R or C). Given a compact topological space K and a continuous map f : K → K n , we denote bythe maximum norm of f on K. The corresponding space of continuous functions on K will be endowed with the C 0 topology associated to · 0,K . Given a topological surface N, we denote by bN the (possibly non-connected) 1dimensional topological manifold determined by its boundary points. Given a subset A ⊂ N, we denote by A • and A the interior and the closure of A in N , respectively. Open connected subsets of N \ bN will be called domains of N , and those proper connected Proof. We begin with the following assertion.Claim 5.7. There exists a connected I-admissible subsetŜ ⊂ N such that RŜ = R S and CŜ ⊃ C S ; that is to say,Ŝ is constructed by adding a finite family of Jordan arcs to S.Proof. If S is connected chooseŜ = S.Assume that S is not connected. We distinguish the following two cases. (See Remark 3.1 and Def. 2.2 and 2.3 for notation.) Case 1. S is a connected subset of the non-orientable Riemann surface N . In this situation, any tubular neighborhood of S is an orientable surface. Then, take any Jordan arc γ ⊂ N with end points in bR S and otherwise disjoint from S , such that any tubular neighborhood

show abstract

Complete Non-Orientable Minimal Surfaces in ℝ³ and Asymptotic Behavior

Abstract: In this paper we give new existence results for complete non-orientable minimal surfaces in R 3 with prescribed topology and asymptotic behavior.Mathematics Subject Classification (2010) 49Q05.

Cited by 3 publications

References 14 publications

Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves: Figure. 5.1.

Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves: Figure. 5.1.

New Complex Analytic Methods in the Study of Non-Orientable Minimal Surfaces in ℝⁿ

Approximation theory for nonorientable minimal surfaces and applications

Contact Info

Product

Resources

About

Complete Non-Orientable Minimal Surfaces in ℝ3 and Asymptotic Behavior

Abstract: In this paper we give new existence results for complete non-orientable minimal surfaces in R 3 with prescribed topology and asymptotic behavior.Mathematics Subject Classification (2010) 49Q05.

Cited by 3 publications

References 14 publications

Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves: Figure. 5.1.

Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves: Figure. 5.1.

New Complex Analytic Methods in the Study of Non-Orientable Minimal Surfaces in ℝⁿ

Approximation theory for nonorientable minimal surfaces and applications

Contact Info

Product

Resources

About

Complete Non-Orientable Minimal Surfaces in ℝ³ and Asymptotic Behavior