## Limits of harmonic maps and crowned hyperbolic surfaces

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## Abstract:

We consider harmonic diffeomorphisms to a fixed hyperbolic target $Y$ from a family of domain Riemann surfaces degenerating along a Teichmüller ray. We use the work of Minsky to show that there is a limiting harmonic map from the conformal limit of the Teichmüller ray to a crowned hyperbolic surface. The target surface is the metric completion of the complement of a geodesic lamination on $Y$. The conformal limit is obtained by attaching half-planes and cylinders to the critical graph of the holomorphic quadratic differential determining the ray. As an application, we provide a new proof of the existence of harmonic maps from any punctured Riemann surface to a given crowned hyperbolic target of the same topological type.## References

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## Additional Information

**Subhojoy Gupta**- Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore 560012, India
- MR Author ID: 1001472
- Email: subhojoy@iisc.ac.in
- Received by editor(s): May 15, 2018
- Received by editor(s) in revised form: December 6, 2018
- Published electronically: June 10, 2019
- Additional Notes: The author thanks the SERB, DST (Grant No. MT/2017/000706) and the Infosys Foundation for its support.
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**372**(2019), 7573-7596 - MSC (2010): Primary 30F60, 57M50, 58E20
- DOI: https://doi.org/10.1090/tran/7777
- MathSciNet review: 4029674