Second order geometric flows on foliated manifolds

TL;DR

This paper establishes short-term existence and uniqueness for second order geometric flows on foliated manifolds, including known flows like transverse Ricci and Sasaki-Ricci flows, and introduces a transverse Kähler-Ricci flow.

Contribution

It provides a unified existence and uniqueness framework for various transverse geometric flows on foliated manifolds, extending classical results to the foliated setting.

Findings

Proved short time existence and uniqueness for transverse second order flows.

Included flows such as transverse Ricci, Sasaki-Ricci, and Sasaki J-flow.

Introduced a transverse Kähler-Ricci flow adapting classical results.

Abstract

We prove a general result about the short time existence and uniqueness of second order geometric flows transverse to a Riemannian foliation on a compact manifold. Our result includes some flows already existing in literature, as the transverse Ricci flow, the Sasaki-Ricci flow and the Sasaki J-flow and motivates the study of other evolution equations. We also introduce a transverse version of the Kaehler-Ricci flow adapting some classical results to the foliated case.