Hypercritical deformed Hermitian-Yang-Mills equation

TL;DR

This paper investigates the hypercritical deformed Hermitian-Yang-Mills equation on compact Kähler manifolds, establishing conditions related to the $\\mathcal{J}$-functional that guarantee solutions exist.

Contribution

It introduces the concepts of coerciveness and properness of the $\\mathcal{J}$-functional and proves their equivalence to the existence of solutions for the hypercritical deformed Hermitian-Yang-Mills equation.

Findings

Coerciveness of the $\\mathcal{J}$-functional implies existence of solutions.

Properness of the $\\mathcal{J}$-functional is equivalent to coerciveness.

Existence of solutions is characterized by these functional properties.

Abstract

In this work, we study the deformed Hermitian-Yang-Mills equation on compact K\"ahler manifold. We introduce the notions of coerciveness and properness of the $\mathcal{J}$-functional on the space of almost calibrated $(1,1)$-forms and show that they are both equivalent to the existence of solutions to the hypercritical deformed Hermitian-Yang-Mills equation.