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arXiv · 2111.14688

Isometries of the Space of Sasaki Potentials

Thomas Franzinetti
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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.

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Thomas Franzinetti. 2021-11-29. Isometries of the Space of Sasaki Potentials. https://arxiv.org/abs/2111.14688

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