The homogeneous generalized Ricci flow

TL;DR

This paper introduces a new framework inspired by bracket flow to analyze the generalized Ricci flow on Lie group quotients, establishing global existence results, defining generalized Ricci solitons, and providing formulas for curvature in this context.

Contribution

It develops a novel bracket flow framework for the generalized Ricci flow on Lie group quotients, proving global existence on solvmanifolds and classifying solitons in low dimensions.

Findings

Global existence of solutions on solvmanifolds in arbitrary dimensions.

Definition and classification of generalized Ricci solitons on nilmanifolds.

New formula for generalized Ricci curvature using a moment map.

Abstract

We develop a framework inspired by Lauret's "bracket flow" to study the generalized Ricci flow, as introduced by Streets, on discrete quotients of Lie groups. As a first application, we establish global existence on solvmanifolds in arbitrary dimensions, a result which is new even for the pluriclosed flow. We also define a notion of generalized Ricci soliton on exact Courant algebroids that is geometrically meaningful and allows for non-trivial expanding examples. On nilmanifolds, we show that these solitons arise as rescaled limits of the generalized Ricci flow, provided the initial metrics have "harmonic torsion", and we classify them in low dimensions. Finally, we provide a new formula for the generalized Ricci curvature of invariant generalized metrics in terms of a moment map for the action of a non-reductive real Lie group.