Journal of Differential Geometry

Minimal surfaces in the round three-sphere by doubling the equatorial two-sphere, I

Nikolaos Kapouleas

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Abstract

We construct closed embedded minimal surfaces in the round three-sphere $\mathbb{S}^3 (1)$, resembling two parallel copies of the equatorial two-sphere $\mathbb{S}^2_{\mathrm{eq}}$, joined by small catenoidal bridges symmetrically arranged either along two parallel circles of $\mathbb{S}^2_{\mathrm{eq}}$, or along the equatorial circle and the poles. To carry out these constructions we refine and reorganize the doubling methodology in ways which we expect to apply also to further constructions. In particular, we introduce what we call “linearized doubling”, which is an intermediate step where singular solutions to the linearized equation are constructed subject to appropriate linear and nonlinear conditions. Linearized doubling provides a systematic approach for dealing with the obstructions involved and also understanding in detail the regions further away from the catenoidal bridges.

Article information

Source
J. Differential Geom., Volume 106, Number 3 (2017), 393-449.

Dates
Received: 30 August 2014
First available in Project Euclid: 15 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1500084022

Digital Object Identifier
doi:10.4310/jdg/1500084022

Mathematical Reviews number (MathSciNet)
MR3680553

Zentralblatt MATH identifier
06846956

Citation

Kapouleas, Nikolaos. Minimal surfaces in the round three-sphere by doubling the equatorial two-sphere, I. J. Differential Geom. 106 (2017), no. 3, 393--449. doi:10.4310/jdg/1500084022. https://projecteuclid.org/euclid.jdg/1500084022


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