Minimal surfaces in the Riemannian product of surfaces

Nikos Georgiou; Brendan Guilfoyle

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arXiv · 2412.06618

Minimal surfaces in the Riemannian product of surfaces

Nikos Georgiou·Brendan Guilfoyle

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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.

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Nikos Georgiou, Brendan Guilfoyle. 2024-12-09. Minimal surfaces in the Riemannian product of surfaces. https://arxiv.org/abs/2412.06618

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