The Metric Induced By The Robin Function (memoirs Of The American Mathematical Society)

This paper continues Yamaguchi's earlier work on the Robin function for bounded domains in [bold]C[italic superscript]n. Yamaguchi showed that if the domain [italic]D is smoothly bounded and pseudoconvex, then the Robin function and its logarithm are both real analytic strongly pseudoconvex exhaustions of the domain. It follows that they may be used as potentials to define Kähler metrics on the domain. In this paper, the authors study the properties of these Kähler metrics, concentrating on the question of completeness. By using an affine scaling technique to blow up the domain at points near the boundary (the scaling constant grows roughly as the inverse of the distance to the boundary), the authors study in some detail the behaviour of the Robin function and the induced metric near the boundary of the domain. They show that if the domain is either strongly pseudoconvex or geometrically convex, then the metric associated to the logarithm of the Robin function is complete. They also conjecture that this is true in general for any smoothly bounded pseudoconvex domain Reveals an interesting connection between classical (Newtonian) potential theory on R2n and the theory of several complex variables on pseudoconvex domains in Cn. The authors bring together many results concerning the Robin function *L associated to the R2n Laplace operator on a pseudoconvex domain in Cn.