Kähler-Einstein fillings

Abstract

We show that on an open bounded smooth strongly pseudoconvex subset of C n , there exists a Kähler-Einstein metric with positive Einstein constant, such that the metric restricted to the Levi distribution of the boundary is conformal to the Levi form. To achieve this, we solve an associated complex Monge-Ampère equation with Dirichlet boundary condition. We also prove uniqueness under some more assumptions on the open set.

The volume of Kähler–Einstein Fano varieties and convex bodies

“…More generally, the proof of the previous theorem shows that it is enough to assume that φ is a continuous solution in the sense of pluripotential theory. As shown in [4] γ < (n + 1) is a sufficient condition for existence of such weak solutions for any pseudoconvex domain (by [30] any such continuous solution is in fact smooth in the interior of Ω). Hence γ = n + 1 appears to be a critical parameter for the equations 1.3 on Ω in the presence of an S 1 −action as above.…”

Monge–Ampére iteration

“…which is the first inequality in equation (18). Lemma 3.4 implies f (0) ≥ τ /λ(A), which yields the second inequality in equation (18).…”

“…which is the first inequality in equation (18). Lemma 3.4 implies f (0) ≥ τ /λ(A), which yields the second inequality in equation (18). Lemma 3.3 implies inf{ ϕ i } = ϕ i (0) and A ϕ * i dλ = −τ , so we can apply Lemma 3.6 to ϕ i to prove substep 1.…”

“…The Ricci iteration was introduced by Rubinstein [30,31,32] and has been since studied by a number of authors [3,13,18,20,27,33]. On Kähler manifolds the Ricci iteration is a sequence of complex Monge-Ampère equations.…”

Poincaré- and Sobolev- Type Inequalities for Complex m-Hessian Equations

“…For the complex n-Hessian operator with p = 1, inequality (1.3) was proved by Berman and Berndtsson in [15] (see also [28]). Later their result was generalized by the authors to the case when p is any positive number, and when Ω is a n-hyperconvex domain in C n , or a compact Kähler manifold [3]).…”