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# 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

DOI: https://dx.doi.org/10.4310/CAG.2016.v24.n2.a7

#### Authors

#### Abstract

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