[13] Variation of the Green function and theory of the p-valued functions

Abstract

Let E be a compact set in the plane with transfinite diameter d(E) > 0, and let Ω denote the unbounded component of its complement. Let g(z, ζ) be Green’s function of Ω with pole at ζ. For any specified point z ₀ ∈ Ω, the perturbation ${z}^{{\ast}} = z +\varepsilon {e}^{i\theta }{(z - z_{0})}^{-1}$ is analytic and univalent outside each neighborhood of z ₀, provided that ɛ > 0 is sufficiently small. Let E ^∗ denote the image of E, and let g ^∗(z ^∗, ζ ^∗) be Green’s function of the corresponding domain Ω^∗. In the paper (Schiffer, C. R. Acad. Sci. Paris, 209, 980–982, 1939), Schiffer introduces the important variational formula

$$\displaystyle{{g}^{{\ast}}(z,\infty ) = g(z,\infty )+\varepsilon \,\text{Re}\left \{{e}^{i\theta }\left [p^{\prime}(z_{ 0},z)p^{\prime}(z_{0},\infty ) - p^{\prime}(z,\infty ){(z - z_{0})}^{-1}\right ]\right \}+O{(\varepsilon }^{2})\,,\ z \in \Omega \,,}$$

where p(z, ζ) is an analytic function with real part g(z, ζ). The proof relies on an interesting representation of Green’s function g(z, ∞) in terms of the Fekete points of E, points that maximize the product used to define the n-diameter d _n (E). Schiffer attributes this beautiful relation to Fekete and says that “the proof will appear soon,” but in fact Fekete’s paper was never published. However, the relation is implicit in a result of Leja (Math. Ann., 111, 501–504, 1935), at least for connected sets E, so that Ω is simply connected and Green’s function has the form g(z, ∞) = log|φ(z)|, where φ maps Ω conformally onto |w| > 1 with φ(∞) = ∞ (cf. Schiffer, Bull. Soc. Math. France, 68, 158–176, 1940).