Optimal bounds for the Kobayashi distance near $\mathcal C^2$-smooth boundary points

TL;DR

This paper extends the known optimal bounds for the Kobayashi distance from smooth strongly pseudoconvex boundary points to more general boundary regularity classes, including $ ext{C}^{2}$ and $ ext{C}^{1,1}$, and also considers non-semipositive boundary points.

Contribution

It generalizes existing bounds for the Kobayashi distance to broader boundary regularity classes and boundary types, including non-semipositive points.

Findings

Optimal bounds hold for $ ext{C}^{2}$-smooth strongly pseudoconvex points.

Upper bounds are extended to $ ext{C}^{1,1}$-smooth cases.

Bounds are provided near non-semipositive boundary points.

Abstract

It is shown that the optimal upper and lower bounds for the Kobayashi distance near $\mathcal C^{2,\alpha}$-smooth strongly pseudoconvex boundary points obtained in L. Kosinski, N. Nikolov, A.Y. Okten: "Precise estimates of invariant distances on strongly pseudoconvex domains", Adv. Math. 478 (2025), 110388, remain true in the general $\mathcal C^2$ strongly pseudoconvex setting. In fact, the upper bound is extended to the general $\mathcal C^{1,1}$-smooth case. We also give upper and lower bounds for the Kobayashi distance near non-semipositive boundary points.