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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Some remarks on deformations of minimal surfaces


Authors: Harold Rosenberg and Éric Toubiana
Journal: Trans. Amer. Math. Soc. 295 (1986), 491-499
MSC: Primary 53A10
MathSciNet review: 833693
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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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DOI: http://dx.doi.org/10.1090/S0002-9947-1986-0833693-2
PII: S 0002-9947(1986)0833693-2
Article copyright: © Copyright 1986 American Mathematical Society