The mixed scalar curvature flow and harmonic foliations

Abstract

We introduce and study the flow of metrics on a foliated Riemannian manifold $(M,g)$, whose velocity along the orthogonal distribution is proportional to the mixed scalar curvature, $\Sc_{\,\rm mix}$. The flow is used to examine the question: When a foliation admits a metric with a given property of $\Sc_{\,\rm mix}$ (e.g., positive or negative)\/? We observe that the flow preserves harmonicity of foliations and yields the Burgers type equation along the leaves for the mean curvature vector $H$ of orthogonal distribution. If $H$ is leaf-wise conservative, then its potential obeys the non-linear heat equation $\dt u=n\Delta_\calf\,u +(n\beta_{\mathcal D}+\Phi)\,u+\Psi^\calf_1 u^{-1}-\Psi^\calf_2 u^{-3}$ with a leaf-wise constant $\Phi$ and known functions $\beta_{\mathcal D}\ge0$ and $\Psi^\calf_i\ge0$. We study the asymptotic behavior of its solutions and prove that under certain conditions (in terms of spectral parameters of leaf-wise Schr\"{o}dinger operator $\mathcal{H}_\calf=-\Delta_\calf -\beta_{\mathcal D}\id$) there exists a unique global solution $g_t$, whose $\Sc_{\rm mix}$ converges exponentially as $t\to\infty$ to a leaf-wise constant. The metrics are smooth on $M$ when all leaves are compact and have finite holonomy group. Hence, in certain cases, there exist ${\mathcal D}$-conformal to $g$ metrics, whose $\Sc_{\rm mix}$ is negative or positive.