Construction of minimal annuli in PSL2 via a variational method

TL;DR

This paper constructs complete minimal annuli in PSL(2) using a variational approach, providing new examples and curvature estimates despite the space's lack of symmetry.

Contribution

It introduces a variational method to construct minimal annuli in PSL(2), including a one-periodic family of examples, overcoming symmetry challenges.

Findings

Constructed complete embedded minimal annuli asymptotic to vertical planes.

Established curvature estimates independent of surface index.

Proved existence of a one-periodic family of Riemann's type examples.

Abstract

We construct complete, embedded minimal annuli asymptotic to vertical planes in the Riemannian 3-manifold PSL. The boundary of these annuli consists of 4 vertical lines at infinity. They are constructed by taking the limit of a sequence of compact minimal annuli. The compactness is obtained from an estimate of curvature which uses foliations by minimal surfaces. This estimate is independent of the index of the surface. We also prove the existence of a one-periodic family of Riemann's type examples. The difficulty of the construction comes from the lack of symmetry of the ambient space PSL.