Variation of geodesic length functions over Teichm\"uller space

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.