Alexandrov curvature of Kaehler curves

TL;DR

This paper investigates the intrinsic geometry of one-dimensional complex spaces with Kähler metrics, establishing that an upper bound on Gaussian curvature implies a bound on Alexandrov curvature.

Contribution

It demonstrates that Gaussian curvature bounds on Kähler curves translate into Alexandrov curvature bounds for the intrinsic metric, linking complex geometry with metric geometry.

Findings

Intrinsic metric has curvature at most K in the sense of Alexandrov.

Gaussian curvature bounds imply Alexandrov curvature bounds.

Connects complex geometric properties with metric curvature notions.

Abstract

We study the intrinsic geometry of a one-dimensional complex space provided with a Kaehler metric in the sense of Grauert. We show that if K is an upper bound for the Gaussian curvature on the regular locus, then the intrinsic metric has curvature at most K in the sense of Alexandrov.