Generalized cross curvature flow

TL;DR

This paper introduces a generalized cross curvature flow for compact 3-manifolds, establishing short-time existence and uniqueness, and provides an example demonstrating the flow's application.

Contribution

It extends the cross curvature flow framework to a broader class of flows on 3-manifolds, with proven existence, uniqueness, and illustrative examples.

Findings

Proved short-time existence and uniqueness of the flow.

Constructed explicit examples of the flow on manifolds.

Extended the applicability of cross curvature flow to new settings.

Abstract

In this paper, for a given compact 3-manifold with an initial Riemannian metric and a symmetric tensor, we establish the short-time existence and uniqueness theorem for extension of cross curvature flow. We give an example of this flow on manifolds.