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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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Some remarks on deformations of minimal surfaces
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by Harold Rosenberg and Éric Toubiana PDF
Trans. Amer. Math. Soc. 295 (1986), 491-499 Request permission

Abstract:

We consider complete minimal surfaces (c.m.s.’s) in ${R^3}$ and their deformations. ${M_1}$ is an $\varepsilon$-deformation of ${M_0}$ if ${M_1}$ is a graph over ${M_0}$ in an $\varepsilon$ tubular neighborhood of ${M_0}$ and ${M_1}$ is $\varepsilon \;{C^1}$-close to ${M_0}$. A minimal surface $M$ is isolated if all c.m.s.’s which are sufficiently small deformations of $M$ are congruent to $M$. In this paper we construct an example of a nonisolated c.m.s. It is modelled on a $4$-punctured sphere and is of finite total curvature. On the other hand, we prove that a c.m.s. discovered by Meeks and Jorge, modelled on the sphere punctured at the fourth roots of unity, is isolated.
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Additional Information
  • © Copyright 1986 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 295 (1986), 491-499
  • MSC: Primary 53A10
  • DOI: https://doi.org/10.1090/S0002-9947-1986-0833693-2
  • MathSciNet review: 833693