Variation of geodesic length functions in families of Kähler-Einstein manifolds and applications to Teichmüller space

Abstract

In the study of Teichmuller spaces the second variation of the logarithm of the geodesic length function plays a central role. So far, it was accessible only in a rather indirect way. We treat the problem directly in the more general framework of the deformation theory of Kahler–Einstein manifolds. For the first variation we arrive at a surprisingly simple formula, which only depends on harmonic Kodaira–Spencer forms. We also compute the second variation in the general case and then apply the result to families of Riemann surfaces. Again we obtain a simple formula depending only on the harmonic Beltrami differentials. As a consequence a new proof for the plurisubharmonicity of the geodesic length function on Teichmuller space and its logarithm together with upper estimates follow. The results also apply to the previously not known cases of Teichmuller spaces of weighted punctured Riemann surfaces, where the methods of Kleinian groups are not available. We use our methods from [A-S], where the result was announced.