Hyperbolicity and Schwarz Lemmas in Calibrated Geometry
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| Format: | Preprint |
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2025
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| author | Broder, Kyle Iliashenko, Anton Madnick, Jesse |
| author_facet | Broder, Kyle Iliashenko, Anton Madnick, Jesse |
| contents | This paper has two main objectives. First, for an arbitrary calibrated manifold $(X,ϕ)$, we define notions of $R_ϕ$-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 $R_ϕ$-hyperbolicity implies $ϕ$-hyperbolicity, and give examples showing that the converse fails in general. Moreover, for constant-coefficient, inner Mobius rigid calibrations $ϕ$ in $\mathbb{R}^n$, 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 $R_ϕ$-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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_16313 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hyperbolicity and Schwarz Lemmas in Calibrated Geometry Broder, Kyle Iliashenko, Anton Madnick, Jesse Differential Geometry 53 Differential geometry This paper has two main objectives. First, for an arbitrary calibrated manifold $(X,ϕ)$, we define notions of $R_ϕ$-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 $R_ϕ$-hyperbolicity implies $ϕ$-hyperbolicity, and give examples showing that the converse fails in general. Moreover, for constant-coefficient, inner Mobius rigid calibrations $ϕ$ in $\mathbb{R}^n$, 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 $R_ϕ$-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. |
| title | Hyperbolicity and Schwarz Lemmas in Calibrated Geometry |
| topic | Differential Geometry 53 Differential geometry |
| url | https://arxiv.org/abs/2507.16313 |