Communications in Analysis and Geometry

Volume 24 (2016)

Number 2

Compact embedded minimal surfaces in $\mathbb{S}^2 \times \mathbb{S}^1$

Pages: 409 – 429



José M. Manzano (Department of Mathematics, King’s College London, The Strand, London, United Kingdom)

Julia Plehnert (Discrete Differential Geometry Lab, University of Göttingen, Germany)

Francisco Torralbo (Departamento Ciencias Básicas, Centro Universitario de la Defensa (CUD) San Javier, Santiago de la Rivera, Spain)


We prove that closed surfaces of all topological types, except for the non-orientable odd-genus ones, can be minimally embedded in $\mathbb{S}^2 \times \mathbb{S}^1 r$, for arbitrary radius $r$. We illustrate it by obtaining some periodic minimal surfaces in $\mathbb{S}^2 \times \mathbb{R}$ via conjugate constructions. The resulting surfaces can be seen as the analogy to the Schwarz P-surface in these homogeneous $3$-manifolds.

2010 Mathematics Subject Classification

Primary 53A10. Secondary 53C30.

Published 14 June 2016