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

Abstract

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).

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

“…It was shown by the authors and López [20,25] that complex analytic methods may also be used in the construction of nonorientable minimal surfaces in R n by working on their oriented double-sheeted coverings. In [20,Example 6.1] the reader can find the first known example of a properly embedded minimal Möbius strip in R 4 (see also Section 2.3). Space does not permit us to include these results.…”

Interpolation by conformal minimal surfaces and directed holomorphic curves

“…This representation formula has greatly influenced the study of minimal surfaces in R n by providing powerful tools coming from Complex Analysis in one and several variables. In particular, Runge and Mergelyan theorems for open Riemann surfaces (see Bishop [18] and also [41,37]) and, more recently, the modern Oka Theory (we refer to the monograph by Forstnerič [27] and to the surveys by Lárusson [36], Forstnerič and Lárusson [29], Forstnerič [26], and Kutzschebauch [35]) have been exploited in order to develop a uniform approximation theory for conformal minimal surfaces in the Euclidean spaces which is analogous to the one of holomorphic functions in one complex variable and has found plenty of applications; see [12,15,7,16,20,11,10,28] and the references therein. In this paper we extend some of the methods invented for developing this approximation theory in order to provide also interpolation on closed discrete subsets of the underlying complex structure.…”

Minimal surfaces in minimally convex domains

“…After the completion of this paper, Alarcón, Forstnerič, and López obtained analogues of Theorems 1.1 and 1.9 in the non-orientable framework (see [4]).…”