We present a brief but nearly self-contained proof of a formula for the Weil-Petersson Hessian of the geodesic length of a closed curve (either simple or not simple) on a hyperbolic surface. The formula is the sum of the integrals of two naturally defined positive functions over the geodesic, proving convexity of this functional over Teichmuller space (due to Wolpert (1987)). We then estimate this Hessian from below in terms of local quantities and distance along the geodesic. The formula extends to proper arcs on punctured hyperbolic surfaces, and the estimate to laminations. Wolpert’s result that the Thurston metric is a multiple of the Weil-Petersson metric directly follows on taking a limit of the formula over an appropriate sequence of curves. We give further applications to upper bounds of the Hessian, especially near pinching loci, recover through a geometric argumentWolpert’s result on the convexity of length to the half-power, and give a lower bound for growth of length in terms of twist.

In this addendum we complete the proof of Theorem 7.10 in [1].

In this paper we establish the short-time existence and uniqueness theorem for hyperbolic geometric flow, and prove the nonlinear stability of hyperbolic geometric flow defined on the Euclidean space with dimension larger than 4. Wave equations satisfied by the curvatures are derived. The relations of the hyperbolic geometric flow with the Einstein equations and the Ricci flow are discussed.

Let φ∈C^0∩W_{1,2}(Σ,X) where Σ is a compact Riemann surface, X is a compact locally CAT(1) space, and W_{1,2}(Σ,X) is defined as in Korevaar-Schoen. We use the technique of harmonic replacement to prove that either there exists a harmonic map u: Σ → X homotopic to φ or there exists a nontrivial conformal harmonic map v: S^2 → X. To complete the argument, we prove compactness for energy minimizers and a removable singularity theorem for conformal harmonic maps.

We study the collapsing behavior of the Kähler–Ricci flow on a compact Kähler manifold X admitting a holomorphic submersion inherited from its canonical bundle. We show that the flow metric degenerates at exactly the rate of e^{-t} as predicted by the cohomology information, and so the fibres collapse at the optimal rate. Consequently, it leads to some analytic and geometric extensions to the regular case of works by J. Song and G. Tian. Its applicability to general Calabi–Yau fibrations will also be discussed in local settings.