Nodal resolution of quasiregular curves via bubble trees

Abstract

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.