Variation of geodesic length functions over Teichmüller space
Reynir Axelsson, Georg Schumacher
TL;DR
The paper develops a hyperbolic-geometric, Kähler-framework for the variation of geodesic length functions across holomorphic families of negatively curved Kähler–Einstein manifolds. It derives an explicit second-variation formula involving resolvent-type operators $ (\Box+1)^{-1}$ and $(-\frac{D^2}{dt^2}+2)^{-1}$, together with a cross-term, and proves that $\log \ell(\gamma_s)$ is strictly plurisubharmonic, along with quantitative upper and lower estimates. Specializing to Teichmüller spaces (dimensionality one in the base) yields a simplified expression in terms of harmonic Beltrami differentials, linking the Weil–Petersson metric to geodesic-length data. The results extend beyond uniformization to conical hyperbolic metrics on weighted punctured Riemann surfaces (weights $\ge 1/2$), providing an intrinsic, group-theory-free approach to plurisubharmonicity and exhaustion properties in these moduli problems.
Abstract
In a family of compact, canonically polarized, complex manifolds equipped with Kähler-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 Teichmüller spaces of weighted punctured Riemann surfaces, where the methods of Kleinian groups are not available.