Geodesics in the space of K\"ahler metrics

TL;DR

This paper investigates the geometric structure of the space of K"ahler metrics on a compact manifold, demonstrating that smooth geodesics generally do not connect arbitrary points in this infinite-dimensional space.

Contribution

It provides a rigorous analysis showing that smooth geodesics do not always exist between points in the space of K"ahler metrics, clarifying the geometric limitations of this space.

Findings

Smooth geodesics do not always exist between points in the space.

The space of K"ahler metrics has a complex geometric structure.

The question of connecting points with smooth geodesics is generally negative.

Abstract

Let (X,\omega) be a compact K\"ahler manifold. As discovered in the late 1980s by Mabuchi, the set H_0 of K\"ahler forms cohomologous to \omega has the natural structure of an infinite dimensional Riemannian manifold. We address the question whether any two points in H_0 can be connected by a smooth geodesic, and show that the answer, in general, is "no".