Isometries of the Space of Sasaki Potentials

TL;DR

This paper explores the isometries of the space of Sasaki potentials, revealing differences from Kähler cases and constructing Sasaki isospectral structures through affine Mabuchi isometries.

Contribution

It demonstrates that unlike Kähler manifolds, Sasaki manifolds can have isometric potential spaces without being diffeomorphic, and introduces new isospectral structures via Mabuchi isometries.

Findings

Sasaki manifolds can have isometric potential spaces without diffeomorphism.

Regular Sasaki manifolds with isometric potential spaces share the same universal cover.

Existence of affine Mabuchi isometries leads to Sasaki isospectral structures.

Abstract

Given any two K\"ahler manifolds $X_1$ and $X_2$, L. Lempert recently proved that if their spaces of K\"ahler potentials are isometric with respect to the Mabuchi metric, then $X_1$ and $X_2$ must be diffeomorphic. We prove that this is no longer the case for Sasaki manifolds. Then, considering regular Sasaki manifolds $M_1$ and $M_2$, we prove that if the spaces of potentials are isometric, then $M_1$ and $M_2$ must have, among others, the same universal covering space. Finally, getting rid of the regularity assumption on $M_1$ and $M_2$, we investigate the consequences of the existence of affine Mabuchi isometries: this leads to a family of Sasaki isospectral structures.