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

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

The objective of those lecture notes is to supply an creation to the speculation of advanced Monge–Ampère operators (definition, regularity matters, geometric houses of strategies, approximation) on compact Kähler manifolds (with or with no boundary).
These operators are of crucial use in different primary difficulties of complicated differential geometry (Kähler–Einstein equation, distinctiveness of continuing scalar curvature metrics), advanced research and dynamics. the subjects lined comprise, the Dirichlet challenge (after Bedford–Taylor), Monge–Ampère foliations and laminated currents, polynomial hulls and Perron envelopes with out analytic constitution, a self-contained presentation of Krylov regularity effects, a modernized facts of the Calabi–Yau theorem (after Yau and Kolodziej), an creation to endless dimensional riemannian geometry, geometric buildings on areas of Kähler metrics (after Mabuchi, Semmes and Donaldson), generalizations of the regularity concept of Caffarelli–Kohn–Nirenberg–Spruck (after Guan, Chen and Blocki) and Bergman approximation of geodesics (after Phong–Sturm and Berndtsson).

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

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For the general case where Σ is merely C 1 we refer the reader to the “second theorem of support” in [Dem93]. A real-analytic Levi flat hypersurface in C2 is locally of the form (z) = 0 in some holomorphic system of coordinates (z, w). We work locally so Σ writes as (z) = 0. Put z = x+iy. Then since x = 0 on supp (T ) and T has measure coefficients, we infer that xT = 0. Since T is ∂ and ∂ closed, we infer that ∂(xT ) = dz ∧ T = 0 and ∂(xT ) = dz ∧ T = 0. Now if χ is a test function which is constant along the leaves, that is, which depends only on z along supp (T ), we infer that χT is closed.

For the reverse inclusion, note that u > 0 in C2 \(B∪X) so that X ˆ What remains to be proved is thus that u(p) > 0 whenever p ∈ B \ X. ˆ There exists a psh function on C2 such that v < 0 on X and Let p ∈ B \ X. v(p) > 0. Fix δ such that v < 0 on Φ ≤ δ, and M such that v ≤ M on ∂B. If 0 < ε < δ/M , we infer that εv ≤ Φ on ∂B. From the definition of u as an upper envelope, we deduce that u ≥ εv on B. In particular u(p) > 0. 17 is “no” in the C 1,1 setting. Regularity C 1,1 is important in pluripotential theory for it is the regularity of solutions to the homogeneous Monge–Amp`ere equation with smooth boundary data.

Then since x = 0 on supp (T ) and T has measure coefficients, we infer that xT = 0. Since T is ∂ and ∂ closed, we infer that ∂(xT ) = dz ∧ T = 0 and ∂(xT ) = dz ∧ T = 0. Now if χ is a test function which is constant along the leaves, that is, which depends only on z along supp (T ), we infer that χT is closed. 5. We now prove that the potentials of uniformly laminar currents are maximal. 11 Let L = {Lα }α∈K ⊂ Ω be a lamination by disjoint graphs, Lα = {(z, w) ∈ D × C / w = fα (z)}. Fix a probability measure μ on K ⊂ C and consider T = Tμ the corresponding uniformly laminar current.

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