Robin Functions for Complex Manifolds and Applications

In [20] and [9] we analyzed the second variation of the Robin function associated to a smooth variation of domains in C for n ≥ 2; i.e., D = ∪t∈B(t, D(t)) ⊂ B×Cn is a variation of domains D(t) in C each containing a fixed point z0 and with ∂D(t) of class C ∞ for t ∈ B := {t ∈ C : |t| < ρ}. For such t and for z ∈ D(t) we let g(t, z) be the R-Green function for the domain D(t) with pole at z0; i.e., g(t, z) is harmonic in D(t)\{z0}, g(t, z) = 0 for z ∈ ∂D(t), and g(t, z)− 1 ||z−z0|| is harmonic near z0. We call