Maximal vorticity of sections of the orthonormal frame bundle via calibrations

Início: 11/08/2023 17:00

Término: 11/08/2023 18:00

Palestrante: Marcos Salvai (FAMAF, Universidad Nacional de Córdoba, Argentina)

E-mail do Palestrante: marcos.salvai@gmail.com

Resumo: Let M be an oriented three dimensional Riemannian manifold. We define a notion of vorticity of local sections of the bundle SO(M) -> M of all its positively oriented orthonormal tangent frames. When M is a space form, we relate the concept to a suitable invariant split pseudo-Riemannian metric on Iso_o (M) equiv SO(M): A local section has positive vorticity if and only if it determines a space-like submanifold. In the Euclidean case we find explicit homologically volume maximizing sections using a split special Lagrangian calibration. We introduce the concept of optimal vorticity and give an optimal screwed global section for the three-sphere. We prove that it is also homologically volume maximizing (now using a common one-point split calibration). Besides, we show that no optimal section can exist in the Euclidean and hyperbolic cases. M. Salvai, A split special Lagrangian calibration associated with frame vorticity, accepted for publication in Adv. Calc. Var.