Abstract
In a family of compact, canonically polarized, complex manifolds equipped
with K"ahler-Einstein metrics the first variation of the lengths of closed
geodesics was previously shown in by the authors in arXiv:0808.3741v2 to be
the geodesic integral of the harmonic Kodaira-Spencer form. We compute the
second variation. For one dimensional fibers we arrive at a formula that only
depends upon the harmonic Beltrami differentials. As an application a new proof
for the plurisubharmonicity of the geodesic length function and its logarithm
(with new upper and lower estimates) follows, which also applies to the
previously not known cases of Teichm"uller spaces of weighted punctured
Riemann surfaces, where the methods of Kleinian groups are not available.
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