Geodesic Convexity of Small Neighborhood in the Space of K\"ahler Potentials

TL;DR

The paper proves that in the space of smooth K"ahler potentials, small neighborhoods exist where any two points can be connected by a geodesic with a certain regularity, under specific smoothness conditions.

Contribution

It establishes the geodesic convexity of small neighborhoods in the space of K"ahler potentials with precise regularity bounds, extending understanding of the geometric structure.

Findings

Existence of small neighborhoods with geodesic connectivity

Geodesics have at least C^{k-J} regularity

Results hold for k > 4 and specific J values

Abstract

We show that, given $k> 4$, $0<J<\min\{{1\over 4},{k-4\over 4}\}$, any point in space of non-degenerate smooth K\"ahler potentials has a small neighborhood with respect to $C^k$ norm, s.t. any two points in this neighborhood can be connected by a geodesic of at least $C^{k-J}$ regularity.