The Siciak–Zahariuta extremal function as the envelope of disc functionals

Abstract

We establish disc formulas for the Siciak-Zahariuta extremal function of an arbitrary open subset of complex affine space. This function is also known as the pluricomplex Green function with logarithmic growth or a logarithmic pole at infinity. We extend Lempert’s formula for this function from the convex case to the connected case. Introduction The Siciak-Zahariuta extremal function VX of a subset X of complex affine space C is defined as the supremum of all entire plurisubharmonic functions u of minimal growth with u|X ≤ 0. It is also called the pluricomplex Green function of X with logarithmic growth or a logarithmic pole at infinity (although this is a bit of a misnomer if X is not bounded). A plurisubharmonic function u on C is said to have minimal growth (and belong to the class L) if u− log ‖·‖ is bounded above on C. If X is open and nonempty, then VX ∈ L. More generally, if X is not pluripolar, then the upper semicontinuous regularization V ∗ X of VX is in L, and if X is pluripolar, then V ∗ X =∞. Siciak-Zahariuta extremal functions play a fundamental role in pluripotential theory and have found important applications in approximation theory, complex dynamics, and elsewhere. For a detailed account of the basic theory, see [K, Chapter 5]. For an overview of some recent developments, see [Pl]. The extremal functions of pluripotential theory are usually defined as suprema of classes of plurisubharmonic functions with appropriate properties. The theory of disc functionals, initiated by Poletsky in the late 1980s [P1, PS], offers a different approach to extremal functions, realizing them as envelopes of disc functionals. A disc functional on a complex manifold Y is a map H into R = [−∞,∞] from the set of analytic discs in Y , that is, holomorphic maps from the open unit disc D into Y . We usually restrict ourselves to analytic discs that extend holomorphically to a neighbourhood of the closed unit disc. 2000 Mathematics Subject Classification. Primary: 32U35. The first-named author was supported in part by the Natural Sciences and Engineering Research Council of Canada. First version 22 April 2005. Second, expanded version 6 July 2005. Typeset by AMS-TEX 1 The envelope EH of H is the map Y → R that takes a point x ∈ Y to the infimum of the values H(f) for all analytic discs f in Y with f(0) = x. Disc formulas have been proved for such extremal functions as largest plurisubharmonic minorants, including relative extremal functions, and pluricomplex Green functions of various sorts, and used to establish properties of these functions that had proved difficult to handle via the supremum definition. Some of this work has been devoted to extending to arbitrary complex manifolds results that were first proved for domains in C. See for instance [BS, E, EP, LS1, LS2, LLS, P2, P3, R, RS]. In the convex case, there is a disc formula for the Siciak-Zahariuta extremal function due to Lempert [M, Appendix]. The main motivation for the present work was to generalize Lempert’s formula. Because of the growth condition in the definition of the Siciak-Zahariuta extremal function, we did not see how to fit it into the theory of disc functionals until we realized, from a remark of Guedj and Zeriahi [GZ], that minimal growth is nothing but quasi-plurisubharmonicity with respect to the current of integration along the hyperplane at infinity. This observation is implicit in the proof of Theorem 1, which presents a family of new disc formulas for the Siciak-Zahariuta extremal function of an arbitrary open subset of affine space. Theorem 2 contains more such formulas. Our main result, Theorem 3, establishes Lempert’s formula, in the following slightly modified form, for every connected open subset of affine space. The formula is easily seen to fail for disconnected sets in general. Theorem. The Siciak-Zahariuta extremal function VX of a connected open subset X of C is given by the disc formula