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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

A property of complete minimal surfaces


Authors: Thomas Hasanis and Dimitri Koutroufiotis
Journal: Trans. Amer. Math. Soc. 281 (1984), 833-843
MSC: Primary 53C42; Secondary 53A10
DOI: https://doi.org/10.1090/S0002-9947-1984-0722778-5
MathSciNet review: 722778
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Abstract: If $ M$ is a complete minimal surface in $ {R^n}$, we denote by $ W$ the set of points in $ {R^n}$ that do not lie on any tangent plane of $ M$. By taking a point in $ W$ as origin, the position vector of $ M$ determines a global unit normal vector field $ e$ to $ M$. We prove that if $ e$ is a minimal section, then $ M$ is a plane. In particular, the set of tangent planes of a nonflat complete minimal surface in $ {R^3}$ covers all $ {R^3}$. We also prove a similar result for a complete minimal surface $ M$ in $ {S^3}$, and deduce from it that if the spherical image of $ M$ lies in a closed hemisphere, then $ M$ is a great $ {S^2}$.


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DOI: https://doi.org/10.1090/S0002-9947-1984-0722778-5
Keywords: Minimal surface, tangent plane, minimal section, support function
Article copyright: © Copyright 1984 American Mathematical Society