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

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.