Ricci Flow on Torus Bundles

TL;DR

This paper derives Ricci flow formulas for invariant metrics on principal torus bundles, exploring their applications to Bakry-Émery Ricci flow and Ricci flows on circle bundles over Kähler-Einstein manifolds, yielding explicit solutions.

Contribution

It provides explicit Ricci flow formulas for invariant metrics on principal G-bundles, especially torus bundles, and applies these to study flows on complex and 3-dimensional manifolds.

Findings

Solutions to Ricci flows on Heisenberg groups

Implicit solutions for Ricci flows on Berger 3-spheres

Analysis of Ricci flow behavior on torus and circle bundles

Abstract

In this paper we compute the Ricci flow formulas for invariant metrics on prinicpal $G$-bundles compatible with the connection. Our primary focus is on torus bundles which we use to study a notion of Bakry-\'Emery Ricci flow as well as Ricci flow on circle bundles over K\"ahler-Einstein manifolds. The latter application gives us solutions to Ricci flows on Heisenberg groups and implicit solutions to Ricci flows on Berger 3-spheres and several other 3-dimensional manifolds.