# Isometries in spaces of K\"ahler potentials

## TL;DR

This paper characterizes local isometries in the infinite-dimensional space of K"ahler potentials with Mabuchi's metric, establishing their existence and uniqueness.

## Contribution

It provides a complete characterization and proof of existence and uniqueness of local isometries in the space of K"ahler potentials.

## Key findings

- Local isometries are characterized explicitly.
- Existence and uniqueness of such isometries are proven.
- The results deepen understanding of the geometric structure of K"ahler potential spaces.

## Abstract

The space of K\"ahler potentials in a compact K\"ahler manifold, endowed with Mabuchi's metric, is an infinite dimensional Riemannian manifold. We characterize local isometries between spaces of K\"ahler potentials, and prove existence and uniqueness for such isometries.