Real and complex earthquakes

Šarić, Dragomir

doi:10.1090/S0002-9947-05-03651-2

Real and complex earthquakes

Author: Dragomir Šarić
Journal: Trans. Amer. Math. Soc. 358 (2006), 233-249
MSC (2000): Primary 30F60, 30F45, 32H02, 32G05; Secondary 30C62
DOI: https://doi.org/10.1090/S0002-9947-05-03651-2
Published electronically: February 4, 2005
MathSciNet review: 2171231
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Abstract: We consider (real) earthquakes and, by their extensions, complex earthquakes of the hyperbolic plane $\mathbb {H}^2$. We show that an earthquake restricted to the boundary $S^1$ of $\mathbb {H}^2$ is a quasisymmetric map if and only if its earthquake measure is bounded. Multiplying an earthquake measure by a positive parameter we obtain an earthquake path. Consequently, an earthquake path with a bounded measure is a path in the universal Teichmüller space. We extend the real parameter for a bounded earthquake into the complex parameter with small imaginary part. Such obtained complex earthquake (or bending) is holomorphic in the parameter. Moreover, the restrictions to $S^1$ of a bending with complex parameter of small imaginary part is a holomorphic motion of $S^1$ in the complex plane. In particular, a real earthquake path with bounded earthquake measure is analytic in its parameter.

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