Non-existence of extremal Sasaki metrics via the Berglund-H\"ubsch transpose

TL;DR

This paper demonstrates, using mirror symmetry and K-stability, that certain Sasaki manifolds with large Sasaki cones cannot admit extremal Sasaki metrics, expanding known examples beyond Brieskorn-Pham polynomials.

Contribution

It introduces a new method based on the Berglund-H"ubsch transpose rule to produce examples of Sasaki manifolds with big cones that lack extremal metrics, generalizing previous constructions.

Findings

Examples of Sasaki manifolds with large cones without extremal metrics.

Duality preserves the non-existence of extremal Sasaki metrics.

Some examples have the homotopy type of a sphere or are rational homology spheres.

Abstract

We use the Berglund-H\"ubsch transpose rule from classical mirror symmetry in the context of Sasakian geometry and results on relative K-stability in the Sasaki setting developed by Boyer and van Coevering to exhibit examples of Sasaki manifolds of big Sasaki cones that do not admit any extremal Sasaki metrics at all. Previously, examples with this feature were produced by Boyer and van Coevering for Brieskorn-Pham polynomials or their deformations. Our examples are based on the more general framework of invertible polynomials. In particular, we construct families of links that preserve the emptiness of the extremal Sasaki-Reeb cone via the Berglund-H\"ubsch rule: if the link does not admit extremal Sasaki metrics then its Berglund-H\"ubsch dual preserves this property and moreover this dual admits a representative in its local moduli with a larger Sasaki-Reeb cone which remains obstructed to admitting extremal Sasaki metrics. Some of the examples exhibited here have the homotopy type of a sphere or are rational homology spheres.