Nodal resolution of quasiregular curves via bubble trees
Pekka Pankka, Jonathan Pim
TL;DR
The work establishes a Gromov-type compactness for sequences of $K$-quasiregular $ω$-curves into closed calibrated manifolds with bounded geometry, producing bubble trees and nodal resolutions that realize weak-$⋆$ limits of the calibrated area measures. Central to the method are removability and Hölder regularity results, the notion of asymptotically quasiregular sequences, and a careful nodal surgery combined with renormalization to redistribute mass and eliminate atoms. The main theorem guarantees the existence of a nodal resolution $\widehat F: \widehat X\to N$ alongside a bubble-tree domain $\widehat X$, with convergence understood as local uniform convergence away from bubbling points and weak-$⋆$ convergence of $✶F_k^{*}ω$ to the push-forward measure of the limit. These results yield a normality criterion for families of quasiregular curves and provide multiple geometric interpretations, including Gromov–Hausdorff convergence and pinching maps, thereby extending bubbling analysis to higher-dimensional calibrated settings with robust geometric control.
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 $ω$ on manifold $N$, we associate to a weak-$\star$ limit $μ= \lim_{k \to \infty} \star F_k^*ω$ of measures induced by a sequence $(F_k \colon X\to N)_{k\in \mathbb{N}}$ of $K$-quasiregular $ω$-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 $μ$, that is, $μ= π_*(\star \widehat F^*ω)$, where $π\colon \widehat X\to X$ 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.