A Metric Lower Bound Estimate for Geodesics in the Space of K\"ahler Potentials

TL;DR

This paper provides a lower bound estimate for eigenvalues related to geodesics in the space of Kähler potentials, ensuring non-degeneracy of metrics along certain geodesics.

Contribution

It establishes a positive lower bound for the eigenvalues of solutions to a degenerate complex Monge-Ampère equation, impacting the understanding of geodesics in Kähler geometry.

Findings

Lower bound estimate for the second smallest eigenvalue of the complex Hessian.

Any two close points in the space of Kähler potentials can be connected by a non-degenerate geodesic.

Abstract

In this paper we establish a positive lower bound estimate for the second smallest eigenvalue of the complex Hessian of solutions to a degenerate complex Monge-Amp\`ere equation. As a consequence, we find that in the space of K\"ahler potentials any two points close to each other in $C^2$ norm can be connected by a geodesic along which the associated metrics do not degenerate.