Weak 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, showing that not all points can be connected by geodesics, even generalized ones, highlighting limitations in the space's geometry.

Contribution

It demonstrates the non-existence of geodesics between certain points in the space of K"ahler metrics, extending previous results with new generalized geodesic considerations.

Findings

Not all points in the space can be joined by geodesics.

Generalized geodesics also do not always exist between arbitrary points.

The space's geometric structure has inherent limitations.

Abstract

Given a compact K\"ahler manifold (X,\omega_0), according to Mabuchi, the set of K\"ahler forms cohomologous to \omega_0 has the natural structure of an infinite dimensional Riemannian manifold. We address the question whether points in this space can be joined by a geodesic, and strengthening previous findings of the second author with Vivas, we show that this cannot always be done even with a certain type of generalized geodesics.