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Asymptotics of Higgs bundles in the Hitchin component

Collier, Brian and Li, Qiongling (2017) Asymptotics of Higgs bundles in the Hitchin component. Advances in Mathematics, 307 . pp. 488-558. ISSN 0001-8708. doi:10.1016/j.aim.2016.11.031.

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In this paper we pursue a more geometric approach to compactification of the Hitchin component. Our main motivation is Wolf's harmonic map interpretation of Thurston's compactification of Teichmüller space with measured foliations. Using Hitchin's parameterization of the Hitchin component by holomorphic differentials, we study asymptotics of certain rays of representations. More precisely, along these rays we solve the Hitchin equations asymptotically and use the solution to study the asymptotics of the parallel transport operator of the associated flat connection. The asymptotics of the corresponding family of equivariant harmonic maps to the symmetric space SL(n,R)/SO(n,R) proves a conjecture of Katzarkov, Noll, Pandit and Simpson [17] on the Hitchin WKB problem in our setting.

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Additional Information:© 2016 Elsevier Inc. Received 13 July 2015, Revised 10 November 2016, Accepted 21 November 2016, Available online 28 November 2016. The authors would like to thank their advisors Steve Bradlow and Mike Wolf for encouraging this collaboration and for their instructive comments. We also thank John Loftin and Jakob Blaavand for their many helpful comments, and Andy Sanders and Daniele Alessandrini for numerous enlightening conversations. We also graciously thank the referee for many useful comments and corrections. Both authors acknowledge the support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network). Our collaboration would not have been possible without this support. The second author is supported by the center of excellence grant ‘Center for Quantum Geometry of Moduli Spaces’ from the Danish National Research Foundation (DNRF95).
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Danish National Research FoundationDNRF95
Subject Keywords:Higgs bundles; Hitchin representations; Asymptotics; Harmonic maps; Toda lattice
Record Number:CaltechAUTHORS:20170707-095300017
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Official Citation:Brian Collier, Qiongling Li, Asymptotics of Higgs bundles in the Hitchin component, Advances in Mathematics, Volume 307, 5 February 2017, Pages 488-558, ISSN 0001-8708, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:78840
Deposited By: Ruth Sustaita
Deposited On:07 Jul 2017 17:06
Last Modified:15 Nov 2021 17:43

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