Compact minimal surfaces in the Berger spheres

TL;DR

This paper constructs new compact minimal surfaces in Berger spheres, characterizes surfaces minimal in all Berger spheres, and generates constant mean curvature surfaces in various 3-manifolds using the Daniel correspondence.

Contribution

It introduces novel methods for constructing minimal surfaces in Berger spheres and links these to constant mean curvature surfaces in other geometries.

Findings

Constructed compact minimal surfaces with arbitrary Euler characteristic in Berger spheres.

Identified a family of surfaces minimal in all Berger spheres.

Generated new constant mean curvature surfaces in S^2 x R, H^2 x R, and the Heisenberg group.

Abstract

We construct compact arbitrary Euler characteristic orientable and non-orientable minimal surfaces in the Berger spheres. Besides we show an interesting family of surfaces that are minimal in every Berger sphere, characterizing them by this property. Finally we construct, via the Daniel correspondence, new examples of constant mean curvature surfaces in the products S^2 x R, H^2 x R and in the Heisenberg group with many symmetries.