We undertake a systematic study of the infinitesimal geometry of the Thurston metric, showing that the topology, convex geometry and metric geometry of the tangent and cotangent spheres based at any marked hyperbolic surface representing a point in Teichmüller space can recover the marking and geometry of this marked surface. We then translate the results concerning the infinitesimal structures to global geometric statements for the Thurston metric, most notably deriving rigidity statements for the Thurston metric analogous to the celebrated Royden theorem.

The paper develops an existence theory for solutions of the Abreu equation, which include extremal metrics on toric surfaces. The technique employed is a continuity method, combined with “blow-up” arguments. General existence results are obtained, assuming a hypothesis (the “M-condition”) on the solutions, which is shown to be related to the injectivity radius.

We study the mean curvature evolution of smooth, closed, twoconvex hypersurfaces in Rn+1 for n ≥ 3.Within this framework we effect a reconciliation between the flow with surgeries—recently constructed by Huisken and Sinestrari in [HS3]—and the wellknown weak solution of the level-set flow: we prove that the two solutions agree in an appropriate limit of the surgery parameters and in a precise quantitative sense. Our proof relies on geometric estimates for certain Lp-norms of the mean curvature which are of independent interest even in the setting of classicalmean curvature flow. We additionally show how our construction can be used to pass these estimates to limits and produce regularity results for the weak solution.

In this paper, we provide new necessary and sufficient conditions for the existence of Kähler–Einstein metrics on small deformations of a Fano Kähler–Einstein manifold. We also show that the Weil–Petersson metric can be approximated by the Ricci curvatures of the canonical L^2 metrics on the direct image bundles. In addition, we describe the plurisubharmonicity of the energy functional of harmonic maps on the Kuranishi space of the deformation of compact Kähler–Einstein manifolds of general type.

Let X be a compact complex, not necessarily K¨ahler, manifold of dimension n. We characterize the volume of any holomorphic line bundle L → X as the supremum R of the Monge-Amp`ere masses X Tn ac over all closed positive currents T in the first Chern class of L, where Tac is the absolutely continuous part of T in its Lebesgue decomposition. This result, new in the non-K¨ahler context, can be seen as holomorphic Morse inequalities for the cohomology of high tensor powers of line bundles endowed with arbitrarily singular Hermitian metrics. It gives, in particular, a new bigness criterion for line bundles in terms of existence of singular Hermitian metrics satisfying positivity conditions. The proof is based on the construction of a new regularization for closed (1, 1)-currents with a control of the Monge-Amp`ere masses of the approximating sequence. To this end, we prove a potential-theoretic result in one complex variable and study the growth of multiplier ideal sheaves associated with increasingly singular metrics.