# The isometries of the space of K\"ahler metrics

## TL;DR

This paper characterizes all global isometries of the space of K"ahler metrics on a compact K"ahler manifold, showing they are induced by biholomorphisms or anti-biholomorphisms, and explores the non-existence of symmetries in Mabuchi's metric.

## Contribution

It proves that all global isometries are induced by biholomorphisms or anti-biholomorphisms, and demonstrates the absence of local symmetries in the Mabuchi metric completion.

## Key findings

- All global isometries are induced by biholomorphisms or anti-biholomorphisms.
- No global symmetries exist for Mabuchi's metric.
- Constructs examples of geodesic segments that cannot be extended at one end.

## Abstract

Given a compact K\"ahler manifold, we prove that all global isometries of the space of K\"ahler metrics are induced by biholomorphisms and anti-biholomorphisms of the manifold. In particular, there exist no global symmetries for Mabuchi's metric. Moreover, we show that the Mabuchi completion does not even admit local symmetries. Closely related to these findings, we provide a large class of metric geodesic segments that can not be extended at one end, pointing out the first such examples in the literature.