Limits of mapping packages and tangents of Sobolev maps
Abstract
We show the existence of ultralimits of sequences of Hajlasz-Sobolev mappings fi:Xi→ Yi when Xi converges to a limit space in the pointed measured Gromov (pmG) sense, partially extending recent results in [T. Ikonen and S. Wenger, (2026), arXiv:2603.05246]. We moreover demonstrate that the mappings pmG-converge to the ultralimit in a suitable sense, and obtain Arzela-Ascoli theorems in the context of pmG-convergence. As an application, we establish a version of Preiss's phenomenon for Hajlasz-Sobolev maps f:(X,μ)→ V into arbitrary Banach spaces. Besides extending it to maps into infinite dimensional targets, our result generalizes existing versions of Preiss's phenomenon [G. C. David, Geom. Funct. Anal., 25 (2015)], [N. Gigli, A. Mondino, and T. Rajala, J. Reine Angew. Math., 705 (2015)] by establishing it for pointed measured Gromov-Hausdorff tangents without a doubling assumption.
Related
- A New insight into the Ambrosio--Reshetnyak approach to Sobolev maps
- Lipschitz regularity of Sobolev maps from RCD(K,N) spaces to CAT(0) spaces
- A relaxation to the post-composition approach to metric-valued Sobolev maps
- New insights into the feature maps of Sobolev kernels: application in global sensitivity analysis
- Critical limits of generating functions for face-constrained labelled planar maps
BibTeX
@misc{cakovic2026limits,
title = {{Limits of mapping packages and tangents of Sobolev maps}},
author = {Milica Caković and Elefterios Soultanis},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.13124},
eprint = {2608.13124},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.13124},
}
RIS
TY - UNPB
TI - Limits of mapping packages and tangents of Sobolev maps
AU - Milica Caković
AU - Elefterios Soultanis
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2608.13124
UR - https://arxiv.org/abs/2608.13124
AB - We show the existence of ultralimits of sequences of Hajlasz-Sobolev mappings f_i:X_i→ Y_i when X_i converges to a limit space in the pointed measured Gromov (pmG) sense, partially extending recent results in [T. Ikonen and S. Wenger, (2026), arXiv:2603.05246]. We moreover demonstrate that the mappings pmG-converge to the ultralimit in a suitable sense, and obtain Arzela-Ascoli theorems in the context of pmG-convergence. As an application, we establish a version of Preiss's phenomenon for Hajlasz-Sobolev maps f:(X,μ)→ V into arbitrary Banach spaces. Besides extending it to maps into infinite dimensional targets, our result generalizes existing versions of Preiss's phenomenon [G. C. David, Geom. Funct. Anal., 25 (2015)], [N. Gigli, A. Mondino, and T. Rajala, J. Reine Angew. Math., 705 (2015)] by establishing it for pointed measured Gromov-Hausdorff tangents without a doubling assumption.
ER -