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Interpolation and optimal hitting for complete minimal surfaces with finite total curvature

  • Antonio AlarcónEmail author
  • Ildefonso Castro-Infantes
  • Francisco J. López
Article
  • 38 Downloads

Abstract

We prove that, given a compact Riemann surface \(\Sigma \) and disjoint finite sets \(\varnothing \ne E\subset \Sigma \) and \(\Lambda \subset \Sigma \), every map \(\Lambda \rightarrow \mathbb {R}^3\) extends to a complete conformal minimal immersion \(\Sigma \setminus E\rightarrow \mathbb {R}^3\) with finite total curvature. This result opens the door to study optimal hitting problems in the framework of complete minimal surfaces in \(\mathbb {R}^3\) with finite total curvature. To this respect we provide, for each integer \(r\ge 1\), a set \(A\subset \mathbb {R}^3\) consisting of \(12r+3\) points in an affine plane such that if A is contained in a complete nonflat orientable immersed minimal surface \(X:M\rightarrow \mathbb {R}^3\), then the absolute value of the total curvature of X is greater than \(4\pi r\). 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 \(\mathbb {R}^3\) 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 

Notes

Acknowledgements

The authors were partially supported by the State Research Agency (SRA) and European Regional Development Fund (ERDF) via the Grants Nos. MTM2014-52368-P and MTM2017-89677-P, MICINN, Spain. They wish to thank an anonymous referee for valuable suggestions which led to an improvement of the exposition.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Antonio Alarcón
    • 1
    Email author
  • Ildefonso Castro-Infantes
    • 1
  • Francisco J. López
    • 1
  1. 1.Departamento de Geometría y Topología e Instituto de Matemáticas (IEMath-GR)Universidad de GranadaGranadaSpain

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