Holomorphic vector fields with a barycentric condition

TL;DR

This paper investigates specific holomorphic vector fields that satisfy a barycentric condition involving their flows, contributing to the understanding of their structure and properties in complex analysis.

Contribution

It introduces and analyzes the barycentric property for tuples of holomorphic vector fields, revealing new structural insights and conditions for such fields.

Findings

Characterization of holomorphic vector fields with barycentric property

Conditions under which the sum of their flows equals a scalar multiple of the identity

New structural results for vector fields satisfying the barycentric condition

Abstract

We study the $p$-tuples of holomorphic vector fields $(X_1,X_2,\ldots,X_p)$ satisfying the barycentric property $\displaystyle\sum_k\exp tX_k=p\cdot\mathrm{id}$, where $\exp tX$ denotes the flow of $X$.