2025/07/22 by Kyle Broder, Broder, Kyle, Anton Iliashenko +3 · 4 citations
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Advanced Differential Geometry Research #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2507.16313
This paper has two main objectives. First, for an arbitrary calibrated manifold (X,ϕ), we define notions of Rϕ-hyperbolicity and ϕ-hyperbolicity, which respectively generalize the notions of Kobayashi and Brody hyperbolicity from complex geometry. To make sense of the former, we introduce the "KR ϕ-metric," a decreasing Finsler pseudo-metric that specializes to the Kobayashi-Royden pseudo-metric in the Kahler case. We prove that Rϕ-hyperbolicity implies ϕ-hyperbolicity, and give examples showing that the converse fails in general. Moreover, for constant-coefficient, inner Mobius rigid calibrations ϕ in ℝn, we completely characterize those domains that are ϕ-hyperbolic. Second, we derive a Schwarz lemma for Smith immersions (a.k.a. conformal ϕ-curves) into an arbitrary calibrated manifold (X, ϕ), thereby extending the Schwarz lemma for holomorphic curves into Kahler manifolds. The relevant Bochner formula features the "ϕ-sectional curvature," a new notion that includes both the scalar and holomorphic sectional curvatures as special cases. As an application, we prove that calibrated geometries with ϕ-sectional curvature bounded above by a negative constant are Rϕ-hyperbolic, generalizing the corresponding result from complex geometry. As another application, we calculate the KR ϕ-metric of real, complex, and quaternionic hyperbolic spaces equipped with their natural calibrations.