A Schwarz lemma of harmonic maps into metric spaces

Jie Wang; Jie Wang

doi:10.3934/era.2024276

Electronic Research Archive

2024, Volume 32, Issue 11: 5966-5974. doi: 10.3934/era.2024276

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A Schwarz lemma of harmonic maps into metric spaces

Jie Wang ^,

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

Received: 01 September 2024 Revised: 17 October 2024 Accepted: 29 October 2024 Published: 06 November 2024

We established a Schwarz lemma for harmonic maps from Riemannian manifolds to metric spaces of curvature bounded above in the sense of Alexandrov. We adopted the gradient estimate technique which was based on Zhang-Zhu's maximum principle. In particular, when the domain manifold was a hyperbolic surface, the energy of any conformal harmonic maps into $ \operatorname{CAT}(-1) $ spaces were bounded from above uniformly.
- harmonic maps,
- singular spaces,
- Schwarz lemma,
- maximum principle,
- Alexandrov curveture bound
Citation: Jie Wang. A Schwarz lemma of harmonic maps into metric spaces[J]. Electronic Research Archive, 2024, 32(11): 5966-5974. doi: 10.3934/era.2024276

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Abstract

We established a Schwarz lemma for harmonic maps from Riemannian manifolds to metric spaces of curvature bounded above in the sense of Alexandrov. We adopted the gradient estimate technique which was based on Zhang-Zhu's maximum principle. In particular, when the domain manifold was a hyperbolic surface, the energy of any conformal harmonic maps into $ \operatorname{CAT}(-1) $ spaces were bounded from above uniformly.

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