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

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