Comparison of the real and the complex Green functions, and sharp estimates of the Kobayashi distance

TL;DR

This paper extends and refines estimates of Green functions and Kobayashi distance in complex domains, providing sharper bounds and unifying classical results for smooth, convexifiable domains of finite type.

Contribution

It generalizes previous upper estimates of Green functions to more complex domains and refines classical Kobayashi distance estimates, unifying them for smooth, convexifiable domains.

Findings

Extended upper estimates for Green functions in complex domains.

Provided lower estimates for Green functions.

Refined and unified classical estimates of Kobayashi distance.

Abstract

We extend the upper estimates obtained by M. Carlehed and B.-Y. Chen about the ratio of the classical and pluricomplex Green functions to the case of $\mathcal C^2$-smooth locally $\mathbb C$-convexifiable domains of finite type. We also give some lower estimates. In order to obtain those results, and because it is of independent interest, we refine and unify some classical estimates about the Kobayashi distance and the Lempert function in such domains.