## Real and complex earthquakes

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- by Dragomir Šarić PDF
- Trans. Amer. Math. Soc.
**358**(2006), 233-249 Request permission

## 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.## References

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## Additional Information

**Dragomir Šarić**- Affiliation: Department of Mathematics, The Gradute School and University Center, The City University of New York, 365 Fifth Avenue, New York, New York 10016
- Address at time of publication: Institute for Mathematical Sciences, SUNY Stony Brook, Stony Brook, New York 11794-3660
- Email: saric@math.sunysb.edu
- Received by editor(s): March 1, 2003
- Received by editor(s) in revised form: February 1, 2004
- Published electronically: February 4, 2005
- © Copyright 2005 American Mathematical Society
- 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
- MathSciNet review: 2171231