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.References
- William H. Meeks III, A survey of the geometric results in the classical theory of minimal surfaces, Bol. Soc. Brasil. Mat. 12 (1981), no. 1, 29–86. MR 671473, DOI 10.1007/BF02588319
- Harold Rosenberg, Deformations of complete minimal surfaces, Trans. Amer. Math. Soc. 295 (1986), no. 2, 475–489. MR 833692, DOI 10.1090/S0002-9947-1986-0833692-0
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