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.
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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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Alarcón, A., Castro-Infantes, I. & López, F.J. Interpolation and optimal hitting for complete minimal surfaces with finite total curvature. Calc. Var. 58, 21 (2019). https://doi.org/10.1007/s00526-018-1465-0
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DOI: https://doi.org/10.1007/s00526-018-1465-0