Details
| Original language | English |
|---|---|
| Pages (from-to) | 213-238 |
| Number of pages | 26 |
| Journal | Journal fur die Reine und Angewandte Mathematik |
| Volume | 2024 |
| Issue number | 817 |
| Early online date | 4 Sept 2024 |
| Publication status | Published - 1 Dec 2024 |
Abstract
We prove the local Lipschitz continuity of sub-elliptic harmonic maps between certain singular spaces, more specifically from the n-dimensional Heisenberg group into CAT(0) spaces. Our main theorem establishes that these maps have the desired Lipschitz regularity, extending the Hölder regularity in this setting proven in [Y. Gui, J. Jost and X. Li-Jost, Subelliptic harmonic maps with values in metric spaces of nonpositive curvature, Commun. Math. Res. 38 (2022), no. 4, 516-534] and obtaining same regularity as in [H.-C. Zhang and X.-P. Zhu, Lipschitz continuity of harmonic maps between Alexandrov spaces, Invent. Math. 211 (2018), no. 3, 863-934] for certain sub-Riemannian geometries; see also [N. Gigli, On the regularity of harmonic maps from RCD(K,N) to CAT(0) spaces and related results, preprint 2022, https://arxiv.org/abs/2204.04317; and A. Mondino and D. Semola, Lipschitz continuity and Bochner-Eells-Sampson inequality for harmonic maps from RCD(k,n) spaces to CAT(0) spaces, preprint 2022, https://arxiv.org/abs/2202.01590] for the generalisation to RCD spaces. The present result paves the way for a general regularity theory of sub-elliptic harmonic maps, providing a versatile approach applicable beyond the Heisenberg group.
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
- Mathematics(all)
- Applied Mathematics
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In: Journal fur die Reine und Angewandte Mathematik, Vol. 2024, No. 817, 01.12.2024, p. 213-238.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Lipschitz regularity of sub-elliptic harmonic maps into CAT(0) space
AU - Assimos, Renan
AU - Gui, Yaoting
AU - Jost, Jürgen
N1 - Publisher Copyright: © 2024 Walter de Gruyter GmbH, Berlin/Boston.
PY - 2024/12/1
Y1 - 2024/12/1
N2 - We prove the local Lipschitz continuity of sub-elliptic harmonic maps between certain singular spaces, more specifically from the n-dimensional Heisenberg group into CAT(0) spaces. Our main theorem establishes that these maps have the desired Lipschitz regularity, extending the Hölder regularity in this setting proven in [Y. Gui, J. Jost and X. Li-Jost, Subelliptic harmonic maps with values in metric spaces of nonpositive curvature, Commun. Math. Res. 38 (2022), no. 4, 516-534] and obtaining same regularity as in [H.-C. Zhang and X.-P. Zhu, Lipschitz continuity of harmonic maps between Alexandrov spaces, Invent. Math. 211 (2018), no. 3, 863-934] for certain sub-Riemannian geometries; see also [N. Gigli, On the regularity of harmonic maps from RCD(K,N) to CAT(0) spaces and related results, preprint 2022, https://arxiv.org/abs/2204.04317; and A. Mondino and D. Semola, Lipschitz continuity and Bochner-Eells-Sampson inequality for harmonic maps from RCD(k,n) spaces to CAT(0) spaces, preprint 2022, https://arxiv.org/abs/2202.01590] for the generalisation to RCD spaces. The present result paves the way for a general regularity theory of sub-elliptic harmonic maps, providing a versatile approach applicable beyond the Heisenberg group.
AB - We prove the local Lipschitz continuity of sub-elliptic harmonic maps between certain singular spaces, more specifically from the n-dimensional Heisenberg group into CAT(0) spaces. Our main theorem establishes that these maps have the desired Lipschitz regularity, extending the Hölder regularity in this setting proven in [Y. Gui, J. Jost and X. Li-Jost, Subelliptic harmonic maps with values in metric spaces of nonpositive curvature, Commun. Math. Res. 38 (2022), no. 4, 516-534] and obtaining same regularity as in [H.-C. Zhang and X.-P. Zhu, Lipschitz continuity of harmonic maps between Alexandrov spaces, Invent. Math. 211 (2018), no. 3, 863-934] for certain sub-Riemannian geometries; see also [N. Gigli, On the regularity of harmonic maps from RCD(K,N) to CAT(0) spaces and related results, preprint 2022, https://arxiv.org/abs/2204.04317; and A. Mondino and D. Semola, Lipschitz continuity and Bochner-Eells-Sampson inequality for harmonic maps from RCD(k,n) spaces to CAT(0) spaces, preprint 2022, https://arxiv.org/abs/2202.01590] for the generalisation to RCD spaces. The present result paves the way for a general regularity theory of sub-elliptic harmonic maps, providing a versatile approach applicable beyond the Heisenberg group.
UR - http://www.scopus.com/inward/record.url?scp=85203519933&partnerID=8YFLogxK
U2 - 10.1515/crelle-2024-0066
DO - 10.1515/crelle-2024-0066
M3 - Article
AN - SCOPUS:85203519933
VL - 2024
SP - 213
EP - 238
JO - Journal fur die Reine und Angewandte Mathematik
JF - Journal fur die Reine und Angewandte Mathematik
SN - 0075-4102
IS - 817
ER -