Hyperbolicity and Schwarz Lemmas in Calibrated Geometry

This paper has two main objectives. First, for an arbitrary calibrated manifold $(X,\phi)$, we define notions of $R_\phi$-hyperbolicity and $\phi$-hyperbolicity, which respectively generalize the notions of Kobayashi and Brody hyperbolicity from complex geometry. To make sense of the former, we introduce the "KR $\phi$-metric," a decreasing Finsler pseudo-metric that specializes to the Kobayashi-Royden pseudo-metric in the Kahler case. We prove that $R_\phi$-hyperbolicity implies $\phi$-hyperbolicity, and give examples showing that the converse fails in general. Moreover, for constant-coefficient, inner Mobius rigid calibrations $\phi$ in $\mathbb{R}^n$, we completely characterize those domains that are $\phi$-hyperbolic. Second, we derive a Schwarz lemma for Smith immersions (a.k.a. conformal $\phi$-curves) into an arbitrary calibrated manifold $(X, \phi)$, thereby extending the Schwarz lemma for holomorphic curves into Kahler manifolds. The relevant Bochner formula features the "$\phi$-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 $\phi$-sectional curvature bounded above by a negative constant are $R_\phi$-hyperbolic, generalizing the corresponding result from complex geometry. As another application, we calculate the KR $\phi$-metric of real, complex, and quaternionic hyperbolic spaces equipped with their natural calibrations.