Generalized Ricci flow on nilpotent Lie groups

Abstract

We define solitons for the generalized Ricci flow on an exact Courant algebroid, building on the definitions of [Gar19]. We then define a family of flows for left-invariant Dorfman brackets on an exact Courant algebroid over a simply connected nilpotent Lie group, generalizing the bracket flows for nilpotent Lie brackets in a way that might make this new family of flows useful for the study of generalized geometric flows, such as the generalized Ricci flow. We provide explicit examples of both constructions on the Heisenberg group.

Ricci-Yang-Mills flow on surfaces and pluriclosed flow on elliptic fibrations

“…A Perelman-type F-functional for this was found in [19], and an expander entropy functional was found in [26]. Some recent results in the homogeneous setting have appeared [20], as well as a stability result near Ricci-flat metrics [23]. In [9] it was shown that the equation reduces to Ricci-Yang-Mills flow in the case of a U (1) principal bundle over a Riemann surface.…”

Generalized Ricci Flow

“…It would be interesting to carry out a similar analysis for left-invariant solutions of generalized Ricci flow in other classes of Lie groups, which help us to understand qualitative properties of the equations. The case of nilpotent Lie groups has been studied in [139] (see also [21,59] in the setting of pluriclosed flow in Chapter 9).…”

On the dynamical behaviour of the generalized Ricci flow

“…Recently, the gRF has been related to some geometric flows in Hermitian Geometry, like e.g. the pluriclosed flow and the generalized Kähler Ricci-flow [5,17,21], and it has been studied on nilpotent Lie groups [12]. We refer the reader to [5] for an extensive introduction to this topic.…”