Nodal resolution of quasiregular curves via bubble trees
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915531868602368 |
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| author | Pankka, Pekka Pim, Jonathan |
| author_facet | Pankka, Pekka Pim, Jonathan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_02927 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nodal resolution of quasiregular curves via bubble trees Pankka, Pekka Pim, Jonathan Differential Geometry Complex Variables Symplectic Geometry Primary 30C65, Secondary 32A30, 53C15, 53C57 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. |
| title | Nodal resolution of quasiregular curves via bubble trees |
| topic | Differential Geometry Complex Variables Symplectic Geometry Primary 30C65, Secondary 32A30, 53C15, 53C57 |
| url | https://arxiv.org/abs/2510.02927 |