Constant Gaussian curvature foliations and Schläfli formulae of hyperbolic 3-manifolds (Q6638212)

Given a closed surface \(\Sigma\), a hyperbolic end \(E\) of topological type \(\Sigma \times (0,\infty)\) is a hyperbolic 3-manifold with underlying topological space \(\Sigma \times (0,\infty)\) whose metric completion is obtained by adding to \(E\) a locally concave pleated surface \(\Sigma \times (0,\infty)\). By work of Thurston, the deformation space of complex projective structures on \(\Sigma\) is in one-to-one correspondence with the space of hyperbolic ends homeomorphic to \(\Sigma \times [0,\infty)\) up to isometry isotopic to the identity. Using this correspondence, Thurston proved that a complex projective structure on \(\Sigma\) is uniquely determined by a pair \((m,\mu)\) where \(m\) is the isotopy class of the metric on the pleated boundary of the hyperbolic end \(E\) associated with \(\sigma\) and \(\mu\) is the measured lamination that records the amount of bending of \(\partial E\).\N\N\par In the paper under review, the author studies the geometry of the foliation by constant Gaussian curvature surfaces \((\Sigma_k)_k\) surfaces of a hyperbolic end and he studies its relation with the structures of its boundary at infinity and of its pleated boundary. First, he shows that the Thurston and the Schwarzian parametrizations are the limits of two families of parametrizations of the space of hyperbolic ends defined by Labourie in terms of the geometry of the leaves \(\Sigma_k\) in the paper [\textit{F. Labourie}, J. Lond. Math. Soc., II. Ser. 45, No. 3, 549--565 (1992; Zbl 0767.53011)]. He gives then a new description of the renormalized volume using the constant curvature foliation. He proves a generalization of a reformulation by of Krasnov and Schlenker of McMullen's Kleinian reciprocity theorem in the role of the Schwarzian parametrization is replaced with Labourie's parametrizations [\textit{K. Krasnov} and \textit{J.-M. Schlenker}, Duke Math. J. 150, No. 2, 331--356 (2009; Zbl 1206.30058)]. The author also describes the constant curvature foliation of a hyperbolic end as the integral curve of a time-dependent Hamiltonian vector field on the cotangent space to Teichmüller space, in analogy to the so-called Moncrief flow for constant mean curvature foliations in Lorentzian space-times [\textit{V. Moncrief}, J. Math. Phys. 30, No. 12, 2907--2914 (1989; Zbl 0704.53076)].\N\N In proving his results the author intoduces two families of volume functions for convex co-compact hyperbolic 3-manifolds: the \(W_k\)-volumes, related to the notion of \(W\)-volume introduced in the paper by Krasnov and Schlenker mentioned above, and the \(V_k^*\)-volumes, related the notion of dual volume introduced by the same authors in the paper [\textit{K. Krasnov} and \textit{J.-M. Schlenker}, Duke Math. J. 150, No. 2, 331--356 (2009; Zbl 1206.30058)]. For both families, he proves a Schläfli-type variation formula, involving the extremal length, in the case of \(W_k\), and the hyperbolic length functions in the case of \(V_k^*\). He also describes a simple way to compute the renormalized volume of a convex co-compact hyperbolic manifold using the volumes \(W_k\).