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Optimal boundary regularity of proper harmonic maps between asymptotically hyperbolic spaces

Part of: Variational problems in infinite-dimensional spaces Representations of solutions Elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  11 September 2025

Jingru Niu
Jingru Niu*
Affiliation:
School of Mathematical Sciences, Beijing Normal University , Beijing 100875, The People’s Republic of China
*
e-mail: 202331130026@mail.bnu.edu.cn
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Abstract

This article studies the optimal boundary regularity of harmonic maps between a class of asymptotically hyperbolic spaces. To be precise, given any smooth boundary map with nowhere vanishing energy density, this article provides an asymptotic expansion formula for harmonic maps under the assumption of $C^1$ up to the boundary.

Keywords

Harmonic mapsboundary regularityasymptotically hyperbolic spaces

MSC classification

Primary: 35J57: Boundary value problems for second-order elliptic systems 35J60: Nonlinear elliptic equations 35C20: Asymptotic expansions 35B65: Smoothness and regularity of solutions 58E20: Harmonic maps , etc.

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Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 38
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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References

Akutagawa, K. and Matsumoto, Y., Proper harmonic maps between asymptotically hyperbolic manifolds . Math. Ann. 364(2016), nos. 3–4, 793–811. https://doi.org/10.1007/s00208-015-1229-5 CrossRefGoogle Scholar
Benoist, Y. and Hulin, D., Harmonic quasi-isometric maps between rank one symmetric spaces . Ann. Math. 185(2017), 895–917. https://doi.org/10.4007/annals.2017.185.3.4 CrossRefGoogle Scholar
Chen, R. Y., Li, S.-Y., and Luo, J., On the asymptotic expansions of the proper harmonic maps between balls in Bergman metrics . J. Geom. Anal. 33(2023), no. 4, Article no. 114, 48 pp. https://doi.org/10.1007/s12220-022-01020-z CrossRefGoogle Scholar
Donnelly, H., Dirichlet problem at infinity for harmonic maps: Rank one symmetric spaces . Trans. Am. Math. Soc. 344(1994), no. 2, 713–735. https://doi.org/10.2307/2154503 CrossRefGoogle Scholar
Donnelly, H., Harmonic maps between rank one symmetric spaces—Regularity at the ideal boundary . Houst. J. Math. 22(1996), no. 1, 73–87.Google Scholar
Donnelly, H., Harmonic maps with noncontact boundary values . Proc. Am. Math. Soc. 127(1999), no. 4, 1231–1241. https://doi.org/10.1090/s0002-9939-99-04627-4 CrossRefGoogle Scholar
Donnelly, H., Asymptotic Dirichlet problem for harmonic maps with bounded image . Geom. Dedicata 91(2002), 1–6.10.1023/A:1016286700896CrossRefGoogle Scholar
Economakis, M., Boundary regularity of the harmonic map problem between asymptotically hyperbolic manifolds, ProQuest LLC, Ann Arbor, MI, 1993.Google Scholar
Fotiadis, A., Harmonic maps between noncompact manifolds . J. Nonlinear Math. Phys. 15(2008), 176–184. https://doi.org/10.2991/jnmp.2008.15.s3.18 CrossRefGoogle Scholar
Graham, C. R., The Dirichlet problem for the Bergman Laplacian. I . Commun. Partial Differ. Equ. 8(1983), no. 5, 433–476. https://doi.org/10.1080/03605308308820275 CrossRefGoogle Scholar
Kim, S. W. and Lee, Y. H., Asymptotic Dirichlet problem for harmonic maps on negatively curved manifolds . J. Korean Math. Soc. 42(2005), no. 3, 543–553. https://doi.org/10.4134/jkms.2005.42.3.543 CrossRefGoogle Scholar
Lee, J. M. and Melrose, R. B., Boundary behaviour of the complex Monge-Ampère equation . Acta Math. 148(1982), 159–192. https://doi.org/10.1007/bf02392727 CrossRefGoogle Scholar
Lemm, M. and Marković, V., Heat flows on hyperbolic spaces . J. Differ. Geom. 108(2018), no. 3, 495–529. https://doi.org/10.4310/jdg/1519959624 CrossRefGoogle Scholar
Leung, M. C., Harmonic maps between asymptotically hyperbolic spaces, ProQuest LLC, Ann Arbor, MI, 1991.Google Scholar
Li, P. and Tam, L.-F., The heat equation and harmonic maps of complete manifolds . Invent. Math. 105(1991), no. 1, 1–46. https://doi.org/10.1007/bf01232256 CrossRefGoogle Scholar
Li, P. and Tam, L.-F., Uniqueness and regularity of proper harmonic maps . Ann. Math. 137(1993), no. 1, 167–201. https://doi.org/10.2307/2946622 CrossRefGoogle Scholar
Li, P. and Tam, L.-F., Uniqueness and regularity of proper harmonic maps. II . Indiana Univ. Math. J. 42(1993), no. 2, 591–635.10.1512/iumj.1993.42.42027CrossRefGoogle Scholar
Li, P. and Wang, J. P., Harmonic rough isometries into Hadamard space . Asian J. Math. 2(1998), no. 3, 419–442. https://doi.org/10.4310/ajm.1998.v2.n3.a2 CrossRefGoogle Scholar
Li, S.-Y. and Ni, L., On the holomorphicity of proper harmonic maps between unit balls with the Bergman metrics . Math. Ann. 316(2000), no. 2, 333–354. https://doi.org/10.1007/s002080050015 CrossRefGoogle Scholar
Li, S.-Y. and Simon, E., On proper harmonic maps between strictly pseudoconvex domains with Kähler metrics of Bergman type . Asian J. Math. 11(2007), no. 2, 251–275. https://doi.org/10.4310/ajm.2007.v11.n2.a5 CrossRefGoogle Scholar
Li, Y. Y., Nguyen, L. and Xiong, J., Regularity of viscosity solutions of the ${\sigma}_k$ -Loewner-Nirenberg problem . Proc. Lond. Math. Soc. (3) 127(2023), no. 1, 1–34. https://doi.org/10.1112/plms.12536 CrossRefGoogle Scholar
Marković, V., Harmonic maps between 3-dimensional hyperbolic spaces . Invent. Math. 199(2015), no. 3, 921–951. https://doi.org/10.1007/s00222-014-0536-x CrossRefGoogle Scholar
Marković, V., Harmonic maps and the Schoen conjecture . J. Am. Math. Soc. 30(2017), no. 3, 799–817. https://doi.org/10.1090/jams/881 CrossRefGoogle Scholar
Mazzeo, R. R., The Hodge cohomology of a conformally compact metric . J. Differ. Geom. 28(1988), no. 2, 309–339. https://doi.org/10.4310/jdg/1214442281 CrossRefGoogle Scholar
Mazzeo, R. R., Elliptic theory of differential edge operators. I . Commun. Partial Differ. Equ. 16(1991), no. 10, 1615–1664. https://doi.org/10.1080/03605309108820815 Google Scholar
Mazzeo, R. R., Regularity for the singular Yamabe problem . Indiana Univ. Math. J. 40(1991), no. 4, 1277–1299.CrossRefGoogle Scholar
Schoen, R., The role of harmonic mappings in rigidity and deformation problems, in complex geometry (Osaka, 1990), Lecture Notes in Pure and Applied Mathematics, 143, Marcel Dekker, Inc., New York, NY, USA, 1990, pp. 179–200.Google Scholar
Wang, J. P., The heat flow and harmonic maps between complete manifolds . J. Geom. Anal. 8(1998), no. 3, 485–514. https://doi.org/10.1007/bf02921799 CrossRefGoogle Scholar