In this paper we study the semigroups of holomorphic maps of a strictly convex domain into itself. In particular, we characterize the semigroups converging, uniformly on compact subsets, to a holomorphic map .
We obtain a solution of the equation ∂u = f as an integral supported only on the bounded convex domain D of Cn, without finite type assumption.
Let X be a convex domain in ℂⁿ and let E be a convex subset of X. The relative extremal function for E in X is the supremum of the class of plurisubharmonic functions v ≤ 0 on X with v ≤ -1 on E. We show that if E is either open or compact, then the sublevel sets of are convex. The proof uses the theory of envelopes of disc functionals and a new result on Blaschke products.
These are the lecture notes of a minicourse given at a winter school in Marseille 2011. The aim of the course was to give an introduction to recent work on the geometry of the space of Kähler metrics associated to an ample line bundle. The emphasis of the course was the role of convexity, both as a motivating example and as a tool.