## Journal of Differential Geometry

### The classification of doubly periodic minimal tori with parallel ends

#### Abstract

Let 𝒦 be the space of properly embedded minimal tori in quotients of ℝ3 by two independent translations, with any fixed (even) number of parallel ends. After an appropriate normalization, we prove that 𝒦 is a 3-dimensional real analytic manifold that reduces to the finite coverings of the examples defined by Karcher, Meeks and Rosenberg in [9, 10, 15]. The degenerate limits of surfaces in 𝒦 are the catenoid, the helicoid and three 1-parameter families of surfaces: the simply and doubly periodic Scherk minimal surfaces and the Riemann minimal examples.

#### Article information

Source
J. Differential Geom., Volume 69, Number 3 (2005), 523-577.

Dates
First available in Project Euclid: 27 July 2005

https://projecteuclid.org/euclid.jdg/1122493998

Digital Object Identifier
doi:10.4310/jdg/1122493998

Mathematical Reviews number (MathSciNet)
MR2170278

Zentralblatt MATH identifier
1094.53007

#### Citation

Pérez, Joaquín; Rodríguez, M. Magdalena; Traizet, Martin. The classification of doubly periodic minimal tori with parallel ends. J. Differential Geom. 69 (2005), no. 3, 523--577. doi:10.4310/jdg/1122493998. https://projecteuclid.org/euclid.jdg/1122493998