Robin functions for complex manifolds and applications

TL;DR

This paper generalizes the second variation formula of Robin functions to complex manifolds with Hermitian metrics, providing a criterion for certain domains to be Stein, thus extending classical potential theory results.

Contribution

It introduces a generalized second variation formula for Robin functions on complex manifolds with a Hermitian metric and a nonnegative function c, enabling new Stein domain criteria.

Findings

Generalized second variation formula for Robin functions in complex manifolds.

Criterion for bounded pseudoconvex domains in homogeneous spaces to be Stein.

Extension of classical potential theory to complex manifolds with additional structures.

Abstract

We prove a generalization of the second variation formula of the Robin function associated to a smooth variation of domains in C^N to the case of the c-Robin function associated to a smooth variation of domains in a complex manifold M equipped with a Hermitian metric and a smooth, nonnegative function c. Our purpose is that, with this added flexibility, we are able to give a criterion for a bounded, smoothly bounded, pseudoconvex domain D in a complex homogeneous space to be Stein.