Complex earthquakes are holomorphic

Earthquakes on compact Riemann surfaces have been studied extensively. They are mappings of the Teichmuller space of a compact Riemann surface to itself. It is a result of Kerckhoff that an earthquake path is a real analytic path in the Teichmuller space of a compact surface. We give a generalization of Kerckhoff's result to the Teichmuller space of any Riemann surface, in fact, to the Universal Teichmuller Space.; We start from a bounded measure on the hyperbolic plane and the corresponding earthquake path parameterized by the positive real numbers. We extend the parameterization to a neighborhood of the real line in the complex plane. The extension is a holomorphic map in the parameter and, for a fixed parameter, it is a one to one map of the unit circle. Hence, the complex earthquake path, with the parameter in the given neighborhood of the real line, is a holomorphic motion of the unit circle. By Slodkowski's theorem, it is extendible to a holomorphic motion of the complex plane. Then, for a fixed positive parameter, the earthquake map is the restriction to the unit circle of a quasiconformal map of the complex plane preserving the unit disk. Thus an earthquake with a bounded measure is quasisymmetric. We also prove that a quasisymmetric earthquake has bounded measure.; The above results taken together show that for an earthquake the following are equivalent: (1) The measure of an earthquake is bounded, (2) An earthquake is quasisymmetric, (3) An earthquake path is a part of a holomorphic motion of the unit circle.; Moreover, an earthquake path with bounded measure is a real analytic path in the Universal Teichmuller Space.