Parabolic flows on almost complex manifolds (Q2414074)

Let $(M,J)$ be a compact almost complex manifold and let $g$ be an almost Hermitian metric on $M$. The associated unique real \((1,1)\)-form $\omega$ with respect to $g$ is referred to an almost Hermitian metric. Given an initial almost Hermitian metric $\omega_0$ on $(M,J)$, the author defines the almost Hermitian flow by \[ (\partial/(\partial t))\omega(t)=\partial\partial_{g(t)}^*\omega(t)+\partial\partial_{g(t)}^*\omega(t)-P(\omega(t))=-\Phi(\omega(t))\omega(0)=\omega_0.\tag{1} \] where $\partial_{g(t)}^*$ and $\partial_{g(t)}^*$ are the $L_2$-adjoint operators with respect to metrics $g(t)$ and $P(\omega(t))$ is one of the Ricci-type curvatures of the Chern curvature. First, the author shows that $\omega\to\Phi(\omega)$ is a strictly elliptic operator for an almost Hermitian metric $\omega$, so the short-time existence. The uniqueness of the solution follows from the standard parabolic theory. Secondly the author defines the almost Hermitian closed flow by \[ ((\partial g(t))/(\partial t))=-S(g(t))-Q^7(g(t))-Q^8(g(t))+BT'(g(t))+Z(g(t)) g(0)=g_0\tag{2} \] where $S$ is one of the Ricci-type curvatures of the Chern curvature, $T$ is the torsion of the Chern connection, $Z$ be an arbitrary local \((1,0)\)-frame around a fixed point $p\in M$ and $B$ denotes the structure coefficients of the Lie bracket. The author shows the following: Theorem. Let $(M,g_0,J)$ be a compact almost Hermitian manifold with fundamental form $\omega_0$. Then the metric $g(t)$ of the solution $\omega(t)$ to (1) starting at $\omega_0$ evolves as (2). The parabolic flow (2) coincides with the flow (1) starting at a pluriclosed metric if $J$ is integrable. Thus the results obtained in this paper constitute in a certain sense an extension to almost Hermitian compact manifolds of the works obtained by \textit{J. Streets} and \textit{G. Tian} [Int. Math. Res. Not. 2010, No. 16, 3101--3133 (2010; Zbl 1198.53077); Geom. Topol. 17, No. 4, 2389--2429 (2013; Zbl 1272.32022)] for Hermitian compact manifolds.