Deforming metrics of foliations (Q352722)

Geometric flows are among the most important tools in modern Riemannian geometry. In the present paper, the authors study evolution equations for Riemannian manifolds equipped with two complementary orthogonal distributions \(D\) and \(D^\perp\), assuming \(D^\perp\) is integrable. They deform the initial metric along \(D\) in the direction of the \(D^\perp\)-divergence of the mean curvature vector \(H\) of \(D\), while keeping the \(D^\perp\) directions unchanged. This flow is studied in relation with the heat flow of the \(1\)-form dual to \(H\). The main result obtained with this technique gives sufficient conditions for convergence of the flow to a limit metric for which \(H\) vanishes identically.