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The mixed scalar curvature flow and harmonic foliations

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 $\dt u=nΔ_\calf\,u +(nβ_{\mathcal D}+Φ)\,u+Ψ^\calf_1 u^{-1}-Ψ^\calf_2 u^{-3}$ with a leaf-wise constant $Φ$ and known functions $β_{\mathcal D}\ge0$ and $Ψ^\calf_i\ge0$. We study the asymptotic behavior of its solutions and prove that under certain conditions (in terms of spectral parameters of leaf-wise Schrödinger operator $\mathcal{H}_\calf=-Δ_\calf -β_{\mathcal D}\id$) 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.