The Metric Induced by the Robin Function

This book 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. Using the technique of variation of domains, the second author proved that, under mild regularity assumptions on the domain, -*L and log (-*L) are strictly plurisubharmonic. In addition to providing a new proof of this result, the authors discuss the asymptotics of the Robin function, the relationship between the Laplacian of the Robin function and the Bergman kernel function, and the completeness of the Kahler metric associated to log(-*L). The book is essentially self-contained and should be accessible to those with knowledge of the basic concepts of several complex variables, classical potential theory, and elementary differential geometry.

Levi-curvature; Smooth variation of domains; Boundary behaviour of the Robin function *L(*x); Proof of Lemma 3.1; Proof of Lemma 3.1, continued; Limiting formulas; Strict plurisubharmonicity of -*L(*x), log(-*L (*x)); The Robin function and the Bergman kernel; Metric induced by Robin function; Strictly pseudoconvex boundary points; Explicit formulas for a half-space; Sufficient conditions for completeness of the *L-metric; An example with J2(*x,a) Ac J J1(*x,a) J2.