The Calabi's metric for the space of Kaehler metrics

TL;DR

This paper explores Calabi's Riemannian metric on the space of Kähler metrics, revealing its geometric structure as part of an infinite-dimensional sphere and providing explicit solutions to geodesic problems.

Contribution

It formalizes Calabi's idea by analyzing the geometry of the space of Kähler metrics and solving key geodesic equations explicitly.

Findings

The space is a portion of an infinite dimensional sphere.

Explicit unique solutions for the geodesic Cauchy and Dirichlet problems.

The space exhibits specific geometric features consistent with Calabi's original ideas.

Abstract

Given any closed Kaehler manifold we define, following an idea by Eugenio Calabi, a Riemannian metric on the space of Kaehler metrics regarded as an infinite dimensional manifold. We prove several geometrical features of the resulting space, some of which we think were already known to Calabi. In particular, the space is a portion of an infinite dimensional sphere and admits explicit unique smooth solutions for the Cauchy and the Dirichlet problems for the geodesic equation.