# Download PDF by Vincent Guedj (auth.), Vincent Guedj (eds.): Complex Monge–Ampère Equations and Geodesics in the Space of

By Vincent Guedj (auth.), Vincent Guedj (eds.)

ISBN-10: 3642236685

ISBN-13: 9783642236686

ISBN-10: 3642236693

ISBN-13: 9783642236693

The goal of those lecture notes is to supply an advent to the idea of complicated Monge–Ampère operators (definition, regularity matters, geometric houses of recommendations, approximation) on compact Kähler manifolds (with or with no boundary).

These operators are of important use in different primary difficulties of advanced differential geometry (Kähler–Einstein equation, specialty of continuous scalar curvature metrics), advanced research and dynamics. the themes lined comprise, the Dirichlet challenge (after Bedford–Taylor), Monge–Ampère foliations and laminated currents, polynomial hulls and Perron envelopes without analytic constitution, a self-contained presentation of Krylov regularity effects, a modernized facts of the Calabi–Yau theorem (after Yau and Kolodziej), an advent to limitless dimensional riemannian geometry, geometric buildings on areas of Kähler metrics (after Mabuchi, Semmes and Donaldson), generalizations of the regularity thought of Caffarelli–Kohn–Nirenberg–Spruck (after Guan, Chen and Blocki) and Bergman approximation of geodesics (after Phong–Sturm and Berndtsson).

Each bankruptcy could be learn independently and relies on a sequence of lectures through R. Berman, Z. Blocki, S. Boucksom, F. Delarue, R. Dujardin, B. Kolev and A. Zeriahi, brought to non-experts. The publication is therefore addressed to any mathematician with a few curiosity in a single of the subsequent fields, advanced differential geometry, advanced research, advanced dynamics, absolutely non-linear PDE's and stochastic analysis.

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It coincides with a continuous function on a set of arbitrary large size with respect to the Monge–Amp`ere measures involved. ✷ We derive from this identity two corollaries which are often called “maximum principle” in the literature. 18 Let Ω Cn be a bounded domain and let u, v be locally bounded psh functions such that lim inf z→∂Ω (u(z) − v(z)) ≥ 0. Then {u

Then (1) {Lα }α∈K extends uniquely to a family of disjoint graphs parameterized by K; (2) the collection L = {Lα }α∈K form a lamination, in the sense that the holonomy map {0} × D → {z} × D is automatically continuous. Laminations by graphs over the unit disk are often called holomorphic motions Proof. Assume without loss of generality that |fα | < 1. Then for α = β, the function h = hα,β := − log(|fα (z) − fβ (z)|/2) is harmonic and positive in the unit disk D. It therefore follows from the Harnack inequality that 1 − |z| h(0) ≤ h(z), for all z ∈ D.

17 Assume u ∈ P SH ∩ C 2 (Ω) is maximal. Is T = ddc u uniformly laminar? The assumption that u is C 2 is natural for ddc u then determines a continuous ﬁeld of complex lines in the tangent bundle, which we can hope could be integrated into a lamination. e. u(z, w) = u(|z| , w), using some properties of solutions to real Monge–Amp`ere equations. The general case remains open. 4 Polynomial Hulls The notion of polynomial hull is a central concept in analysis in several complex variables. We brieﬂy recall its deﬁnition and the connection made by Bremermann with the Dirichlet problem for the complex Monge–Amp`ere operator.

### Complex Monge–Ampère Equations and Geodesics in the Space of Kähler Metrics by Vincent Guedj (auth.), Vincent Guedj (eds.)

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