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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Limits of harmonic maps and crowned hyperbolic surfaces
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by Subhojoy Gupta PDF
Trans. Amer. Math. Soc. 372 (2019), 7573-7596 Request permission

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.
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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