Long geodesics in the space of K\"ahler metrics

TL;DR

This paper investigates the properties of long geodesics in the space of K"ahler metrics, showing their connection to holomorphic vector fields for regular cases and exploring implications for moment map convexity.

Contribution

It establishes that regular geodesics in the space of K"ahler metrics are induced by holomorphic vector fields and extends convexity results for moment maps.

Findings

Regular geodesics are induced by holomorphic vector fields

Results on the derivative of geodesics imply a convexity theorem for moment maps

Open question remains for generalized geodesics

Abstract

We give some remarks on geodesics in the space of K\"ahler metrics that are defined for all time. Such curves are conjecturally induced by holomorphic vector fields, and we show that this is indeed so for regular geodesics, whereas the question for generalized geodesics is still open (as far as we know). We also give a result about the derivative of such geodesics which implies a variant of a theorem of Atiyah and Guillemin-Sternberg on convexity of the image of certain moment maps.