Beiträge zur Theorie der Minimalflächen

Lie, Sophus

doi:10.1007/bf01677141

Cited by 26 publications

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“…Non-orientable minimal surfaces have also attracted considerable interest. Although the study of non-orientable minimal surfaces in R 3 was initiated by Lie back in 1878 (see [Lie78]), the global theory, and in particular the use of complex analytic tools in its development, only began in the eighties of the last century with the seminal paper of Meeks [Mee81]; see also López [Lóp93,Lóp96] and the references therein. Every surface is locally orientable, and hence we can study harmonic maps at least locally as real parts of holomorphic maps.…”

Section: A Summary Of the Main Resultsmentioning

confidence: 99%

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

2020

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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).

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Section: A Summary Of the Main Resultsmentioning

confidence: 99%

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

2020

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“…Non-orientable minimal surfaces in R n . These surfaces appeared in the very origin of Minimal Surface Theory (we refer to the seminal paper by Lie [29] from 1878) and there is a large literature devoted to their study. Conformal non-orientable minimal surfaces in R n for n ≥ 3 are characterized as the images of conformal minimal inmmersions X : M → R n such that X • I = X, where I : M → M is an antiholomorphic involution without fixed points on an open Riemann surface M .…”

Section: Analogous Results For Other Families Of Surfacesmentioning

confidence: 99%

Complete minimal surfaces densely lying in arbitrary domains of ℝn

Alarcón

Castro-Infantes

2017

Geom. Topol.

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In this paper we prove that, given an open Riemann surface M and an integer n ≥ 3, the set of complete conformal minimal immersions M → R n with X(M ) = R n forms a dense subset in the space of all conformal minimal immersions M → R n endowed with the compact-open topology. Moreover, we show that every domain in R n contains complete minimal surfaces which are dense on it and have arbitrary orientable topology (possibly infinite); we also provide such surfaces whose complex structure is any given bordered Riemann surface.Our method of proof can be adapted to give analogous results for non-orientable minimal surfaces in R n (n ≥ 3), complex curves in C n (n ≥ 2), holomorphic null curves in C n (n ≥ 3), and holomorphic Legendrian curves in C 2n+1 (n ∈ N).

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“…There are many classical and modern minimal surfaces in the literature. See Darboux [1,2], Dierkes [3], Fomenko and Tuzhilin [4], Gray, Salamon, and Abbena [5], Nitsche [6], Osserman [7], Spivak [8] for some books, Lie [9], Schwarz [10], Small [11,12], and Weierstrass [13,14] for some papers related to minimal surfaces in Euclidean geometry.…”

Section: Introductionmentioning

confidence: 99%

“…Lie [9] studied the algebraic minimal surfaces and gave a table classifying these surfaces. See also Enneper [15], Güler [16], Nitsche [6], and Ribaucour [17] for details.…”

Section: Introductionmentioning

confidence: 99%