The mixed scalar curvature flow and harmonic foliations

TL;DR

This paper introduces a flow of metrics on foliated Riemannian manifolds driven by mixed scalar curvature, analyzing its effects on harmonic foliations, and establishing conditions for convergence to metrics with prescribed scalar curvature properties.

Contribution

It develops a new geometric flow based on mixed scalar curvature, studies its properties, and proves convergence results for metrics with specific scalar curvature characteristics.

Findings

Flow preserves harmonicity of foliations.

Solutions to the associated nonlinear heat equation converge exponentially.

Existence of metrics with prescribed positive or negative mixed scalar curvature.

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.