On the derivatives of the Lempert functions

TL;DR

This paper proves that under certain conditions, the derivatives of higher order Lempert functions exist and match the Kobayashi metrics, extending Kobayashi's earlier results for taut manifolds.

Contribution

It generalizes Kobayashi's result by establishing the existence and equality of derivatives of higher order Lempert functions with Kobayashi metrics under continuity and positivity conditions.

Findings

Derivatives of higher order Lempert functions exist under specified conditions.

These derivatives are equal to the Kobayashi metrics at the point.

The result extends previous work by Kobayashi to broader classes of manifolds.

Abstract

We show that if the Kobayashi--Royden metric of a complex manifold is continuous and positive at a given point and any non-zero tangent vector, then the "derivatives" of the higher order Lempert functions exist and equal the respective Kobayashi metrics at the point. It is a generalization of a result by M. Kobayashi for taut manifolds.