An Upper Bound on the Critical Volume in a Class of Toric Sasaki-Einstein Manifolds

TL;DR

This paper establishes an upper bound on the critical volume for a broad class of toric Sasaki-Einstein manifolds, linking geometric bounds to the properties of associated toric Calabi-Yau varieties.

Contribution

It introduces a new upper bound on the volume of these manifolds and characterizes cases when this bound is achieved, using both geometric analysis and computational tools.

Findings

Upper bound on critical volume established

Characterization of cases attaining the bound

Use of neural networks and gradient saliency in analysis

Abstract

We prove the existence of an upper bound on critical volume of a large class of toric Sasaki-Einstein manifolds with respect to the first Chern class of the resolutions of the Gorenstein singularities in the corresponding toric Calabi-Yau varieties. We examine the canonical metrics obtained by the Delzant construction on these varieties and characterise cases when the bound is attained. We comment on computational tools used in the investigation, in particular Neural Networks and the gradient saliency method.