NON-EXISTENCE OF 1-ST GAUDUCHON METRIC IN THE CONFORMAL CLASS OF A METRIC ON 6-DIMENSIONAL ALMOST HERMITIAN MANIFOLDS

Abstract

We investigate a semilinear equation and associate each almost Hermitian metric with a unique constant ${\gamma }_k$. Then, we show that there is no 1-st Gauduchon metric in the conformal class of an almost Hermitian metric $\omega$ satisfying that ${\gamma }_1\left(\omega \right)>0$ on a real 6-dimensional compact almost Hermitian manifold. Especially, we find that there is no pluriclosed metric in the conformal class of $\omega$.