Duke Mathematical Journal

On properly embedded minimal surfaces with three ends

Francisco Martín and Matthias Weber

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We classify all complete embedded minimal surfaces in ℝ3; with three ends of genus g and at least 2g+2 symmetries. The surfaces in this class are the Costa-Hoffman-Meeks surfaces that have 4g+4 symmetries in the case of a flat middle end. The proof consists of using the symmetry assumptions to deduce the possible Weierstrass data and then studying the period problems in all cases. To handle the 1-dimensional period problems, we develop a new general method to prove convexity results for period quotients. The 2-dimensional period problems are reduced to the 1-dimensional case by an extremal length argument.

Article information

Duke Math. J., Volume 107, Number 3 (2001), 533-559.

First available in Project Euclid: 5 August 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]


Martín, Francisco; Weber, Matthias. On properly embedded minimal surfaces with three ends. Duke Math. J. 107 (2001), no. 3, 533--559. doi:10.1215/S0012-7094-01-10735-7. https://projecteuclid.org/euclid.dmj/1091737023

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