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Transactions of the American Mathematical Society

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

 

 

Gauss map of minimal surfaces with ramification


Author: Min Ru
Journal: Trans. Amer. Math. Soc. 339 (1993), 751-764
MSC: Primary 53A10
DOI: https://doi.org/10.1090/S0002-9947-1993-1191614-3
MathSciNet review: 1191614
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Abstract: We prove that for any complete minimal surface $ M$ immersed in $ {R^n}$, if in $ C{P^{n - 1}}$ there are $ q > n(n + 1)/2$ hyperplanes $ {H_j}$ in general position such that the Gauss map of $ M$ is ramified over $ {H_j}$ with multiplicity at least $ {e_j}$ for each $ j$ and

$\displaystyle \sum\limits_{j = 1}^q {\left({1 - \frac{{(n - 1)}} {{{e_j}}}} \right) > n(n + 1)/2} $

, then $ M$ must be flat.

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DOI: https://doi.org/10.1090/S0002-9947-1993-1191614-3
Article copyright: © Copyright 1993 American Mathematical Society