arXiv subjects

Jonathan Pim

Publications and source records attributed to Jonathan Pim.

2 recordsLinked to original sources

arXiv2026

A rescaling principle for quasiregular curves with applications to hyperbolicity

We prove a Miniowitz--Zalcman rescaling principle for quasiregular curves into calibrated manifolds. We have two main applications. First, we introduce Brody hyperbolicity adapted to our setting and prove its equivalence to the normality of the family of quasiregular curves from the Euclidean unit ball into the target. When normality holds, we quantify the local modulus of continuity for quasiregular curves using an injectivity radius lower bound and a sectional curvature upper bound of the target. Second, in the special case of conformal curves into closed calibrated manifolds, we prove the equivalence of Kobayashi and Brody hyperbolicity. This answers a question posed by Broder--Iliashenko--Madnick. As an intermediate result, we prove an analogue of Marty's theorem from complex analysis in this setting. Additionally, we construct new examples of non-constant entire quasiregular curves factoring through a Special Lagrangian submanifold of a closed Calabi--Yau manifold, an associative submanifold of a closed $G_2$ manifold, and four-dimensional analogues thereof, providing obstructions to Brody hyperbolicity.

math.DG↗

arXiv2025

Nodal resolution of quasiregular curves via bubble trees

We prove a version of Gromov's compactness theorem for quasiregular curves into calibrated manifolds with bounded geometry. In our main theorem, given an $n$-dimensional calibration $\omega$ on manifold $N$, we associate to a weak-$\star$ limit $\mu = \lim_{k \to \infty} \star F_k^*\omega$ of measures induced by a sequence $(F_k \colon X\to N)_{k\in \mathbb{N}}$ of $K$-quasiregular $\omega$-curves on a nodal manifold $X$, a bubble tree $\widehat X$ over $X$, a sequence of mappings $(\widehat F_\ell \colon X \to N)_{\ell \in \mathbb{N}}$ converging locally uniformly to a quasiregular curve $\widehat F\colon \widehat X\to N$ which realizes the measure $\mu$, that is, $\mu = \pi_*(\star \widehat F^*\omega)$, where $\pi \colon \widehat X\to X$ is the natural projection. We call the sequence $(\widehat F_\ell)_{\ell \in \mathbb{N}}$ a nodal resolution of the sequence $(F_k)_{k\in \mathbb{N}}$. As a corollary we obtain a normality criterion for families of quasiregular curves. Classic interpretations of bubbling via Gromov--Hausdorff convergence and pinching maps also follow.

math.DG↗