@article{ba28f8496f2a48a5959d3127622ca14f,
title = "Convergence of the Weil–Petersson metric on the Teichm{\"u}ller space of bordered Riemann surfaces",
abstract = "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 Teichm{\"u}ller space which is a Hilbert manifold and which is holomorphically included in the standard Teichm{\"u}ller 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 Teichm{\"u}ller space is given by (Formula presented.) harmonic Beltrami differentials. We conclude that this Teichm{\"u}ller space has a finite Weil–Petersson Hermitian metric. Finally, we show that the aforementioned Teichm{\"u}ller space is locally modeled on a space of (Formula presented.) harmonic Beltrami differentials.",
keywords = "(Formula presented.) Beltrami differentials, bordered Riemann surfaces, Gardiner–Schiffer variation, infinitesimally trivial Beltrami differentials, Teichm{\"u}ller theory, Weil–Petersson metric",
author = "David Radnell and Eric Schippers and Wolfgang Staubach",
year = "2016",
month = "6",
day = "14",
doi = "10.1142/S0219199716500255",
language = "English",
volume = "19",
journal = "Communications in Contemporary Mathematics",
issn = "0219-1997",
publisher = "World Scientific Publishing Co.",
number = "01",
}