Prescribing the Mixed Scalar Curvature of a Foliation

Abstract

We introduce the flow of metrics on a foliated Riemannian manifold (M g), whose velocity along the orthogonal (to the foliation $\mathcal{F}$) distribution $\mathcal{D}$ is proportional to the mixed scalar curvature, Scal_mix. The flow preserves harmonicity of foliations and is used to examine the question: When does a foliation admit a metric with a given property of Scal_mix (e.g., positive/negative or constant)? If the mean curvature vector of $\mathcal{D}$ is leaf-wise conservative, then its potential function obeys the nonlinear heat equation $(1/n)\partial _{t}u = \Delta _{\mathcal{F}}\,u + (\beta _{\mathcal{D}} + \Phi /n)u + (\Psi _{1}^{\mathcal{F}}/n){u}^{-1} - (\Psi _{2}^{\mathcal{F}}/n){u}^{-3}$ with a leaf-wise constant $\Phi$ and known functions $\beta _{\mathcal{D}}\geq 0$ and $\Psi _{i}^{\mathcal{F}}\geq 0$. We study the asymptotic behavior of its solutions and prove that under certain conditions (in terms of spectral parameters of Schrödinger operator $\mathcal{H}_{\mathcal{F}} = -\Delta _{\mathcal{F}}-\beta _{\mathcal{D}}\mathrm{id\,}$) the flow of metrics admits a unique global solution, whose Scal_mix converges exponentially to a leaf-wise constant. Hence, in certain cases, there exists a $\mathcal{D}$-conformal to g metric, whose Scal_mix is negative, positive, or negative constant.