Teoría de interpolación por superficies mínimas

Abstract

Interpolation Theory by Minimal Surfaces: An immersed surface in the Euclidean space of dimension n>2 is said to be a minimal surface if it is locally area-minimizing, that is to say, small pieces of it are the ones with least area among all the surfaces with the same boundary. This family of surfaces has been an important topic of research in differential geometry. A crucial fact in minimal surface theory was the discovery of an analytic representation formula proved by Enneper and Weierstrass. The so-called Enneper-Weierstrass representation formula characterizes any minimal surface in the n-dimensional Euclidean space in terms of holomorphic data on a Riemann surface with certain properties. The Enneper-Weierstrass representation formula has greatly influenced the study of minimal surfaces in the Euclidean space by providing powerful tools coming from complex analysis. In particular, Runge-Mergelyan theorem for open Riemann surfaces has been exploited in order to develop a uniform approximation theory for conformal minimal surfaces in Euclidean spaces which is analogous to the one of holomorphic functions in one complex variable. Nevertheless, the true power of the relation between minimal surfaces and complex analysis has been exploited only recently with the use of the powerful complex analytic methods coming from Oka theory. This connection has allowed to prove existence results for minimal surfaces with interesting global properties and prescribing the conformal structure. For a detailed explanation see the survey [AF]. Approximation theory by holomorphic functions is a central topic in complex analysis. It began with the classical Runge Theorem that characterizes topologically the subsets of the complex plane for which any holomorphic function on them may be uniformly approximated by entire functions. This connection has allowed to prove generalizations to the Runge and Mergelyan theorems for the family of conformal minimal immersions, see [AL] for the n=3 dimensional case and [AFL] for the general case n>2. Besides approximation theory, interpolation theory by holomorphic functions is also a central topic in complex analysis. It began in 1885 with the classical Weierstrass interpolation theorem asserting that it is possible to prescribe the values of an entire function on a divergent sequence of points in the Euclidean complex plane. However, interpolation results for conformal minimal immersions have not been treated before. In this thesis we deal with an interpolation problem for the family of conformal minimal immersions providing an analogous to the Weierstrass interpolation theorem for conformal minimal immersions. Concretely we prove that given a closed and discrete subset in an open Riemann surface, every map defined on it extends to a conformal minimal immersion. This theorem is the first result dealing with an interpolation problem for conformal minimal immersions. We deduce this result as a consequence of a much more general result that ensures not only interpolation but also jet-interpolation of given finite order, approximation on holomorphically convex compact subsets, control on the flux, and global properties such as completeness and, under natural assumptions, properness and injectivity. We also prove a Mergelyan-Weierstrass type theorem, that is, both approximation and interpolation, for a family of directed holomorphic immersions which includes null curves: those complex curves directed by the null quadric. It is well known that real and imaginary parts of null curves are conformal minimal immersions whose flux map vanishes everywhere; and conversely, every conformal minimal immersion is locally, on every simply-connected domain, the real part of a null curve. These results appear in [AC2]. Minimal surfaces with total finite curvature has been one of the main focus of interest in the global theory of minimal surfaces in the 3-dimensional Euclidean space. They have the simplest topological, conformal and asymptotic geometry. However, the holomorphic flexibility of the null quadric does not extend to the algebraic category; this is why the construction methods developed for the above results do not provide complete minimal surfaces with finite total curvature. Only in dimension n=3 and exploiting the spinorial representation for minimal surfaces, it was possible to prove a Runge-Mergelyan type uniform approximation theorem for complete minimal surfaces with finite total curvature, see [L]. For this family of surfaces we combine the ideas of [AC2] and [L] to prove analogues to the aforementioned results for conformal minimal immersions with finite total curvature. These results appear in [ACL]. These theorems open the door to a new line of research, namely, the study of optimal hitting problems in the framework of complete minimal surfaces in R3 with finite total curvature. This problem consists on deciding which is the simplest minimal surface that contains a given finite subset in terms of the total finite curvature. That is, determining the surface with smaller finite total curvature (in absolute value) among all the surfaces that contains a fixed finite subset of points. In this line, we shall provide an upper bound on the cardinal of the intersection between any affine line and a minimal surface with finite total curvature not containing the line. With this bound in hand, we prove an existence result for subsets that are against to the families of minimal surfaces with total curvature bounds for some prescribed value, meaning that there do not exist minimal surfaces in this family containing the set. The last results appearing in the thesis are existence results for complete and dense minimal surfaces in the Euclidean space. It is a natural question whether a given domain in the Euclidean space contains complete minimal surfaces which are dense on it; prior to our results, only the whole space was known to enjoy this property. We show a general existence result for complete minimal surfaces which are densely immersed in any given domain of Rn for arbitrary dimension n>2. We provide such surfaces with arbitrary orientable topology and flux map; moreover, if n>4 we give examples with no self-intersections. Furthermore, in the case when the domain is the whole space then we construct such surfaces not only with arbitrary topology but also with arbitrary complex structure. Such a result for any general domain and any conformal structure does not hold true since it is known that a general domain does not contain minimal surfaces with arbitrary complex structure. We also prove that every domain contains complete minimal surfaces which are dense on it and whose complex structure is any given bordered Riemann surface. Both results are presented in [AC1]. Bibliografía/Bibliography [AC1] A. Alarcón and I. Castro-Infantes. Complete minimal surfaces densely lying in arbitrary domains of Rn. Geom. Topol., 22(1):571–590, 2018. [AC2] A. Alarcón and I. Castro-Infantes. Interpolation by conformal minimal surfaces and directed holomorphic curves. Analysis \& PDE, 12(2):561–604, 2019. [ACL] A. Alarcón, I. Castro-Infantes, and F. J. López. Interpolation and optimal hitting for complete minimal surfaces with finite total curvature. Preprint arXiv1712.04727. [AF] A. Alarcón and F. Forstnerič. New complex analytic methods in the theory of minimal surfaces: a survey. J. Aust. Math. Soc., in press. [AFL] A. Alarcón, F. Forstnerič, and F. J. López. Embedded minimal surfaces in Rn. Math. Z., 283(1-2):1–24, 2016. [AL] A. Alarcón and F. J. López. Minimal surfaces in R3 properly projecting into R2. J. Differential Geom., 90(3):351–381, 2012. [L] F. J. López. Uniform approximation by complete minimal surfaces of finite total curvature in R3. Trans. Amer. Math. Soc., 366(12):6201–6227, 2014.