Extrinsic geometric flows on foliated manifolds, I

TL;DR

This paper investigates the evolution of Riemannian metrics on foliated manifolds through extrinsic geometric flows, establishing local existence, uniqueness, and specific examples like extrinsic Ricci flow and applications to surface foliations.

Contribution

It introduces a framework for analyzing extrinsic geometric flows on foliated manifolds, including existence theorems and explicit examples such as extrinsic Newton transformation and Ricci flows.

Findings

Proved local existence and uniqueness of solutions.

Estimated existence time for particular flow cases.

Provided examples and applications to surface foliations.

Abstract

We study deformations of Riemannian metrics on a given manifold equipped with a codimension-one foliation subject to quantities expressed in terms of its second fundamental form. We prove the local existence and uniqueness theorem and estimate the existence time of solutions for some particular cases. The key step of the solution procedure is to find (from a system of quasilinear PDE's) the principal curvatures of the foliation. Examples for extrinsic Newton transformation flow, extrinsic Ricci flow, and applications to foliations on surfaces are given.