Montel's theorem and tautness in calibrated geometry
Abstract
We relate the hyperbolicity of a calibrated manifold $(X, ϕ)$ to the analytic properties of the space of Smith immersions SmIm(Bk, X) from the Poincare k-ball into X. In particular, we establish the following calibrated analogue of a theorem of Royden: if X is ϕ-saturated, then Rϕ- and Kϕ-hyperbolicity coincide, and either implies the equicontinuity of SmIm(Bk, X) with respect to the ϕ-distance. This yields a Montel theorem for compact ϕ-saturated calibrated manifolds as an immediate corollary. Our primary technical tool is a new Schwarz lemma for Smith immersions from Bk into X, which is of independent interest. In a similar spirit, we also prove a calibrated analogue of Kiernan's theorem to the effect that the Kϕ-hyperbolicity of X is almost equivalent to SmIm(Bk, X) being a normal family. Finally, we prove that bounded domains in flat euclidean space are Rϕ-hyperbolic for any calibration ϕ, and we investigate the hyperbolicity of products and discrete quotients.
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BibTeX
@misc{iliashenko2026montels,
title = {{Montel's theorem and tautness in calibrated geometry}},
author = {Anton Iliashenko and Spiro Karigiannis and Jesse Madnick},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.31393},
eprint = {2606.31393},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.31393},
}
RIS
TY - UNPB
TI - Montel's theorem and tautness in calibrated geometry
AU - Anton Iliashenko
AU - Spiro Karigiannis
AU - Jesse Madnick
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2606.31393
UR - https://arxiv.org/abs/2606.31393
AB - We relate the hyperbolicity of a calibrated manifold $(X, ϕ)$ to the analytic properties of the space of Smith immersions SmIm(B^k, X) from the Poincare k-ball into X. In particular, we establish the following calibrated analogue of a theorem of Royden: if X is ϕ-saturated, then R_ϕ- and K_ϕ-hyperbolicity coincide, and either implies the equicontinuity of SmIm(B^k, X) with respect to the ϕ-distance. This yields a Montel theorem for compact ϕ-saturated calibrated manifolds as an immediate corollary. Our primary technical tool is a new Schwarz lemma for Smith immersions from B^k into X, which is of independent interest. In a similar spirit, we also prove a calibrated analogue of Kiernan's theorem to the effect that the K_ϕ-hyperbolicity of X is almost equivalent to SmIm(B^k, X) being a normal family. Finally, we prove that bounded domains in flat euclidean space are R_ϕ-hyperbolic for any calibration ϕ, and we investigate the hyperbolicity of products and discrete quotients.
ER -