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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Gauss map of minimal surfaces with ramification
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Trans. Amer. Math. Soc. 339 (1993), 751-764 Request permission

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 \[ \sum \limits _{j = 1}^q {\left ({1 - \frac {{(n - 1)}} {{{e_j}}}} \right ) > n(n + 1)/2} \] , then $M$ must be flat.
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Additional Information
  • © Copyright 1993 American Mathematical Society
  • 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