The mixed Yamabe problem for foliations

The mixed scalar curvature of a foliated Riemannian manifold, i.e., an averaged mixed sectional curvature, has been considered by several geometers. We explore the Yamabe type problem: to prescribe the leafwise constant mixed scalar curvature for a foliation by a conformal change of the metric in normal directions only. For a harmonic foliation, we derive the leafwise elliptic equation and explore the corresponding nonlinear heat type equation on a closed manifold (leaf). Then we assume that a foliation is defined by an orientable fiber bundle, and use spectral parameters of certain Schrödinger operator to find solution, which is an attractor of the equation.