Minimal surfaces in circle bundles over Riemann surfaces

TL;DR

This paper proves the existence of embedded minimal surfaces in certain circle bundles over Riemann surfaces, which are sections over the complement of finitely many points, under compatible metrics.

Contribution

It establishes the existence of embedded minimal surfaces in circle bundles over Riemann surfaces with even Euler number, extending understanding of minimal surface existence in 3-manifolds.

Findings

Existence of embedded minimal surfaces in specified circle bundles

Minimal surfaces are sections over the complement of finitely many points

Results apply to bundles with even Euler number

Abstract

For a compact 3-manifold $M$ which is a circle bundle over a compact Riemann surface $\Sigma$ with even Euler number $e(M)$, and with a Riemannian metric compatible with the bundle projection, there exists a compact minimal surface $S$ in $M$. $S$ is embedded and is a section of the restriction of the bundle to the complement of a finite number of points in $\Sigma$.