Energy properness and Sasakian-Einstein metrics

TL;DR

This paper explores the relationship between energy functional properness and the existence of Sasakian-Einstein metrics, establishing inequalities and conditions that guarantee their existence in Sasakian geometry.

Contribution

It demonstrates that the existence of Sasakian-Einstein metrics is equivalent to energy properness and proves related inequalities under specific conditions.

Findings

Existence of Sasakian-Einstein metrics implies a Moser-Trudinger type inequality.

A Miyaoka-Yau type inequality is established in Sasakian geometry.

Properness of energy functionals is closely linked to Sasakian-Einstein metrics.

Abstract

In this paper, we show that the existence of Sasakian-Einstein metrics is closely related to the properness of corresponding energy functionals. Under the condition that admitting no nontrivial Hamiltonian holomorphic vector field, we prove that the existence of Sasakian-Einstein metric implies a Moser-Trudinger type inequality. At the end of this paper, we also obtain a Miyaoka-Yau type inequality in Sasakian geometry.