# Convergence of the Weil–Petersson metric on the Teichmüller space of bordered Riemann surfaces

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**Convergence of the Weil–Petersson metric on the Teichmüller space of bordered Riemann surfaces.** / Radnell, David; Schippers, Eric; Staubach, Wolfgang.

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*Communications in Contemporary Mathematics*, vol. 19, no. 01, 1650025. https://doi.org/10.1142/S0219199716500255

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*Communications in Contemporary Mathematics*,

*19*(01), [1650025]. https://doi.org/10.1142/S0219199716500255

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TY - JOUR

T1 - Convergence of the Weil–Petersson metric on the Teichmüller space of bordered Riemann surfaces

AU - Radnell, David

AU - Schippers, Eric

AU - Staubach, Wolfgang

PY - 2016/6/14

Y1 - 2016/6/14

N2 - Consider a Riemann surface of genus (Formula presented.) bordered by (Formula presented.) curves homeomorphic to the unit circle, and assume that (Formula presented.). For such bordered Riemann surfaces, the authors have previously defined a Teichmüller space which is a Hilbert manifold and which is holomorphically included in the standard Teichmüller space. We show that any tangent vector can be represented as the derivative of a holomorphic curve whose representative Beltrami differentials are simultaneously in (Formula presented.) and (Formula presented.), and furthermore that the space of (Formula presented.) differentials in (Formula presented.) decomposes as a direct sum of infinitesimally trivial differentials and (Formula presented.) harmonic (Formula presented.) differentials. Thus the tangent space of this Teichmüller space is given by (Formula presented.) harmonic Beltrami differentials. We conclude that this Teichmüller space has a finite Weil–Petersson Hermitian metric. Finally, we show that the aforementioned Teichmüller space is locally modeled on a space of (Formula presented.) harmonic Beltrami differentials.

AB - Consider a Riemann surface of genus (Formula presented.) bordered by (Formula presented.) curves homeomorphic to the unit circle, and assume that (Formula presented.). For such bordered Riemann surfaces, the authors have previously defined a Teichmüller space which is a Hilbert manifold and which is holomorphically included in the standard Teichmüller space. We show that any tangent vector can be represented as the derivative of a holomorphic curve whose representative Beltrami differentials are simultaneously in (Formula presented.) and (Formula presented.), and furthermore that the space of (Formula presented.) differentials in (Formula presented.) decomposes as a direct sum of infinitesimally trivial differentials and (Formula presented.) harmonic (Formula presented.) differentials. Thus the tangent space of this Teichmüller space is given by (Formula presented.) harmonic Beltrami differentials. We conclude that this Teichmüller space has a finite Weil–Petersson Hermitian metric. Finally, we show that the aforementioned Teichmüller space is locally modeled on a space of (Formula presented.) harmonic Beltrami differentials.

KW - (Formula presented.) Beltrami differentials

KW - bordered Riemann surfaces

KW - Gardiner–Schiffer variation

KW - infinitesimally trivial Beltrami differentials

KW - Teichmüller theory

KW - Weil–Petersson metric

UR - http://www.scopus.com/inward/record.url?scp=84974779249&partnerID=8YFLogxK

U2 - 10.1142/S0219199716500255

DO - 10.1142/S0219199716500255

M3 - Article

VL - 19

JO - Communications in Contemporary Mathematics

JF - Communications in Contemporary Mathematics

SN - 0219-1997

IS - 01

M1 - 1650025

ER -

ID: 6488108