Deforming metrics of foliations

TL;DR

This paper studies the deformation of geometric structures on manifolds with orthogonal distributions, introducing a flow based on mean curvature and analyzing its long-term behavior and applications to prescribing mean curvature of foliations.

Contribution

It introduces the Extrinsic Geometric Flow for distributions, proves its existence and convergence, and applies it to problems of prescribing mean curvature in foliations.

Findings

Existence and convergence of the Extrinsic Geometric Flow.

Application to prescribing mean curvature vector fields.

Examples include harmonic, umbilical foliations, and double-twisted product metrics.

Abstract

Our results concern geometry of a manifold endowed with a pair of complementary orthogonal distributions (plane fields) and a time-dependent Riemannian metric. The work begins with formulae concerning deformations of geometric quantities as the Riemannian metric varies conformally along one of the distributions. Then we introduce the Extrinsic Geometric Flow depending on the mean curvature vector field of the distribution, and show existence/uniquenes and convergence of a solution as $t\to\infty$, when the complementary distribution is integrable with compact leaves. We apply the method to the problem of prescribing mean curvature vector field of a foliation, and give examples for harmonic and umbilical foliations and for the double-twisted product metrics, including the codimension-one case.