Interpolation and optimal hitting for complete minimal surfaces with finite total curvature
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Calculus of Variations
Interpolation and optimal hitting for complete minimal surfaces with finite total curvature Antonio Alarcón1 · Ildefonso Castro-Infantes1 · Francisco J. López1 Received: 28 December 2017 / Accepted: 7 December 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract We prove that, given a compact Riemann surface and disjoint finite sets ∅ = E ⊂ and ⊂ , every map → R3 extends to a complete conformal minimal immersion \ E → R3 with finite total curvature. This result opens the door to study optimal hitting problems in the framework of complete minimal surfaces in R3 with finite total curvature. To this respect we provide, for each integer r ≥ 1, a set A ⊂ R3 consisting of 12r + 3 points in an affine plane such that if A is contained in a complete nonflat orientable immersed minimal surface X : M → R3 , then the absolute value of the total curvature of X is greater than 4πr . In order to prove this result we obtain an upper bound for the number of intersections of a complete immersed minimal surface of finite total curvature in R3 with a straight line not contained in it, in terms of the total curvature and the Euler characteristic of the surface. Mathematics Subject Classification 53A10 · 52C42 · 30D30 · 32E30
1 Introduction Complete minimal surfaces of finite total curvature have been one of the main focus of interest in the global theory of minimal surfaces in R3 ; we refer for instance to [8,16,18,23] for background on the topic. This subject is intimately related to the one of meromorphic functions and 1-forms on compact Riemann surfaces. Indeed, if M is an open Riemann surface and X : M → R3 is a complete conformal minimal immersion with finite total curvature, then there are a compact Riemann surface and a finite set ∅ = E ⊂ such that M is
Communicated by C. DeLellis.
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Antonio Alarcón [email protected] Ildefonso Castro-Infantes [email protected] Francisco J. López [email protected]
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Departamento de Geometría y Topología e Instituto de Matemáticas (IEMath-GR), Universidad de Granada, Campus de Fuentenueva s/n, 18071 Granada, Spain 0123456789().: V,-vol
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biholomorphic to \ E and the exterior derivative d X of X : \ E → R3 , which coincides with its (1, 0)-part ∂ X since X is harmonic, is holomorphic and extends meromorphically to with effective poles at all points in E (i.e., it can not be finite at any end). In particular, the Gauss map \ E → S2 of X extends conformally to and, up to composing with the stereographic projection, is a meromorphic function on . Any orientable complete minimal surface in R3 of finite total curvature comes in this way (see Osserman [18]). Polynomial Interpolation is a fundamental subject in mapping theory. Given disjoint finite sets E = ∅ and in a compact Riemann surface , the classical Riemann-Roch theorem enables to prescribe the values on of a meromorphic function on that is holomorphic in \ E. Our first main result is an analogue for complete minim
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