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
T2 - Communications in Contemporary Mathematics
JF - Communications in Contemporary Mathematics
SN - 0219-1997
IS - 01
M1 - 1650025
ER -