Zbigniew Błocki

On regularization of plurisubharmonic functions on manifolds

We study the question of when a γ-plurisubharmonic function on a complex manifold, where γ is a fixed (1, 1)-form, can be approximated by a decreasing sequence of smooth 7-plurisubharmonic functions. We show in particular that it is always possible in the compact Kahler case.

The Bergman metric and the pluricomplex Green function

We improve a lower bound for the Bergman distance in smooth pseudoconvex domains due to Diederich and Ohsawa. As the main tool we use the pluricomplex Green function and an L 2 -estimate for the ∂-operator of Donnelly and Fefferman.

Interior regularity of the complex Monge-Ampère equation in convex domains

The theory of fully nonlinear elliptic operators of second order can be applied to the operator (det(uij )) 1/n. It follows in particular that if u is strictly psh and C2,α for some α ∈ (0,1), then det(uij ) ∈ C implies u ∈ Ck+2,β , where k = 1,2, . . . , and β ∈ (0,1) (see, e.g., [9, Lemma 17.16]). Therefore, to prove Theorem A, it is enough to show existence of a solution that is C2,α in every ′ , where α ∈ (0,1) depends on ′. We obtain this assuming only that ψ1/n is positive and Lipschitz in (see Theorem 4.1). In a special case of a polydisc, we also allow nonzero boundary values.

On uniform estimate in Calabi-Yau theorem

We show that the uniform estimate in the Calabi-Yau theorem easily follows from the local stability of the complex Monge-Ampère equation.

Some estimates for the Bergman kernel and metric in terms of logarithmic capacity

For a bounded domain Ω on the plane we show the inequality cΩ(z) 2 ≤ 2πKΩ(z), z ∈ Ω, where cΩ(z) is the logarithmic capacity of the complement C \ Ω with respect to z and KΩ is the Bergman kernel. We thus improve a constant in an estimate due to T. Ohsawa but fall short of the inequality cΩ(z) 2 ≤ πKΩ(z) conjectured by N. Suita. The main tool we use is a comparison, due to B. Berndtsson, of the kernels for the weighted complex Laplacian and the Green function. We also show a similar estimate for the Bergman metric and analogous results in several variables. §

A Lower Bound for the Bergman Kernel and the Bourgain-Milman Inequality

For pseudoconvex domains in \(\mathbb{C}^{n}\) we prove a sharp lower bound for the Bergman kernel in terms of volume of sublevel sets of the pluricomplex Green function. For n = 1 it gives in particular another proof of the Suita conjecture. If \(\Omega \) is convex then by Lempert’s theory the estimate takes the form \(K_{\Omega }(z) \geq 1/\lambda _{2n}(I_{\Omega }(z))\), where \(I_{\Omega }(z)\) is the Kobayashi indicatrix at z. One can use this to simplify Nazarov’s proof of the Bourgain-Milman inequality from convex analysis. Possible further applications of Lempert’s theory in this area are also discussed.

The Calabi–Yau Theorem

This lecture, based on a course given by the author at Toulouse in January 2005, surveys the proof of Yau’s celebrated solution to the Calabi conjecture, through the solvability of inhomogeneous complex Monge– Ampere equations on compact Kahler manifolds.

Smooth exhaustion functions in convex domains

We show that in every bounded convex domain in Rn there exists a smooth convex exhaustion function ψ such that the product of all eigenvalues of the matrix (∂ψ/∂xj∂xk) is ≥ 1. Moreover, if the domain is strictly convex, then ψ can be chosen so that every eigenvalue is ≥

The Complex Monge–Ampère Equation in Kähler Geometry

We will discuss two main cases where the complex Monge–Ampere equation (CMA) is used in Kaehler geometry: the Calabi–Yau theorem which boils down to solving nondegenerate CMA on a compact manifold without boundary and Donaldson’s problem of existence of geodesics in Mabuchi’s space of Kaehler metrics which is equivalent to solving homogeneous CMA on a manifold with boundary. At first, we will introduce basic notions of Kaehler geometry, then derive the equations corresponding to geometric problems, discuss the continuity method which reduces solving such an equation to a priori estimates, and present some of those estimates. We shall also briefly discuss such geometric problems as Kaehler–Einstein metrics and more general metrics of constant scalar curvature.

ON THE SUITA CONJECTURE FOR SOME CONVEX ELLIPSOIDS IN C 2

It was recently shown that for a convex domain Ω in and w ∈ Ω, the function , where is the Bergman kernel on the diagonal and the Kobayashi indicatrix, satisfies . While the lower bound is optimal, not much more is known about the upper bound. In general, it is quite difficult to compute even numerically, and the largest value of it obtained so far is 1.010182… . In this article, we present precise, although rather complicated, formulas for the ellipsoids Ω = {|z1|2m + |z2|2 < 1} (with m ≥ 1/2) and all w, as well as for Ω = {|z1| + |z2| < 1} and w on the diagonal. The Bergman kernel for those ellipsoids was already known; the main point is to compute the volume of the Kobayashi indicatrix. It turns out that in the second case, the function is not C3, 1.