## On Quadratic Differential Metrics on a Closed Riemann Surface

J. Math. Sci. Univ. Tokyo
Vol. 21 (2014), No. 2, Page 221–234.

Sun, Zongliang
On Quadratic Differential Metrics on a Closed Riemann Surface
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Abstract:
We study properties of the space of quadratic differential metrics on a closed Riemann surface of genus $g \geq 2.$ First, we introduce a natural metric on this space defined via length distortions which is proper and compact. Second, we study topological properties of this space and show equivalence of various convergences. Besides, we relate the preceding metric to another metric which is obtained via global Lipschitz constants.

Keywords: Holomorphic quadratic differential, length distortion, Lipschitz constant, metric topology.

Mathematics Subject Classification (2010): Primary 30F45; Secondary 46B20, 51M25.
Mathematical Reviews Number: MR3288809