Minimal surfaces in the Riemannian product of surfaces

TL;DR

This paper investigates minimal surfaces in general Riemannian products of surfaces, revealing geometric restrictions, classifications of totally geodesic surfaces, and bounds on minimal surface areas based on curvature conditions.

Contribution

It extends the study of minimal surfaces to more general Riemannian products, providing new classifications and bounds based on curvature assumptions.

Findings

Totally geodesic surfaces are locally slices or products of geodesics.

No minimal 2-spheres exist when factor curvatures are negative.

Minimal 2-tori are Lagrangian with respect to product symplectic structures.

Abstract

Minimal surfaces in the Riemannian product of surfaces of constant curvature have been considered recently, particularly as these products arise as spaces of oriented geodesics of 3-dimensional space-forms. This papers considers more general Riemannian products of surfaces and explores geometric and topological restrictions that arise for minimal surfaces. We show that generically, a totally geodesic surface in a Riemannian product is locally either a slice or a product of geodesics. If the Gauss curvatures of the factors are negative, it is proven that there are no minimal 2-spheres, while minimal 2-tori are Lagrangian with respect to both product symplectic structures. If the surfaces have non-zero bounded curvatures, we establish a sharp lower bound on the area of minimal 2-spheres and explore the properties of the Gauss and normal curvatures of general compact minimal surfaces.