Free boundary minimal surfaces in the unit three-ball via desingularization of the critical catenoid and the equatorial disc

Author:

Kapouleas Nikolaos1,Li Martin Man-chun2

Affiliation:

1. Department of Mathematics , Brown University , Box 1917, 151 Thayer Street , Providence , RI 02912 , USA

2. Department of Mathematics , The Chinese University of Hong Kong , Shatin , Hong Kong , Hong Kong

Abstract

Abstract We construct a new family of high genus examples of free boundary minimal surfaces in the Euclidean unit 3-ball by desingularizing the intersection of a coaxial pair of a critical catenoid and an equatorial disc. The surfaces are constructed by singular perturbation methods and have three boundary components. They are the free boundary analogue of the Costa–Hoffman–Meeks surfaces and the surfaces constructed by Kapouleas by desingularizing coaxial catenoids and planes. It is plausible that the minimal surfaces we constructed here are the same as the ones obtained recently by Ketover by using the min-max method.

Publisher

Walter de Gruyter GmbH

Subject

Applied Mathematics,General Mathematics

Reference62 articles.

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2. E. G. Björling, In integrationem aequationis derivatarum partialum superfici, cujus in puncto unoquoque principales ambo radii curvedinis aequales sunt signoque contrario, Arch. Math. Phys. 4 (1844), 290–315.

3. J. Chen, A. Fraser and C. Pang, Minimal immersions of compact bordered Riemann surfaces with free boundary, Trans. Amer. Math. Soc. 367 (2015), no. 4, 2487–2507.

4. J. Choe and M. Soret, New minimal surfaces in 𝕊3\mathbb{S}^{3} desingularizing the Clifford tori, Math. Ann. 364 (2016), no. 3–4, 763–776.

5. C. J. Costa, Example of a complete minimal immersion in 𝐑3{\mathbf{R}}^{3} of genus one and three embedded ends, Bol. Soc. Brasil. Mat. 15 (1984), no. 1–2, 47–54.

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