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Quadratic differentials, half-plane structures, and harmonic maps to trees

Gupta, Subhojoy and Wolf, Michael (2016) Quadratic differentials, half-plane structures, and harmonic maps to trees. In: COMMENTARII MATHEMATICI HELVETICI, 91 (2). pp. 317-356.

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Official URL: http://dx.doi.org/10.4171/CMH/388


Let (Sigma, p) be a pointed Riemann surface and k >= 1 an integer. We parametrize the space of meromorphic quadratic differentials on Sigma with a pole of order k + 2 at p, having a connected critical graph and an induced metric composed of k Euclidean half-planes. The parameters form a finite-dimensional space L similar or equal to R-k x S-1 that describe a model singular-flat metric around the puncture with respect to a choice of coordinate chart. This generalizes an important theorem of Strebel, and associates, to each point in T-g,T- 1 x L, a unique metric spine of the surface that is a ribbon-graph with k infinite-length edges to p. The proofs study and relate the singular-flat geometry of the quadratic differential, and the infinite-energy harmonic map from Sigma \textbackslash p to a k-pronged tree, having the same Hopf differential.

Item Type: Journal Article
Additional Information: Copy right for this article belongs to the EUROPEAN MATHEMATICAL SOC, PUBLISHING HOUSE, E T H-ZENTRUM SEW A27, SCHEUCHZERSTRASSE 70, CH-8092 ZURICH, SWITZERLAND
Keywords: Meromorphic differentials; singular-flat metrics; Riemann surfaces; harmonic maps
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Date Deposited: 11 Jun 2016 09:53
Last Modified: 11 Jun 2016 09:53
URI: http://eprints.iisc.ac.in/id/eprint/53930

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