A rigid theorem for deformed Hermitian-Yang-Mills equation

Abstract

In this paper, we study the deformed Hermitian-Yang-Mills equation on compact Kähler manifold with non-negative orthogonal bisectional curvature. We prove that the curvatures of deformed Hermitian-Yang-Mills metrics are parallel with respect to the background metric if there exists a positive constant $C$ such that $-\frac{1}{C}\omega <\sqrt{-1}F<C\omega$. We also study the self-shrinker over ${\mathbb{C}}^n$ to the corresponding parabolic flow. We prove that the self-shrinker over ${\mathbb{C}}^n$ is a quadratic polynomial function. We also show the similar rigid theorem for the J-equations and the self-shrinkers over ${\mathbb{C}}^n$ to J-flow.