The algebraic structure of geometric flows in two dimensions

TL;DR

This paper explores the algebraic structure underlying two-dimensional geometric flows, unifying Ricci and Calabi flows through Toda field equations and continual Lie algebras, and extends the framework to higher-order and Kahler manifold flows.

Contribution

It introduces a novel algebraic framework connecting Ricci and Calabi flows via Toda equations and continual Lie algebras, enabling formal solutions and hierarchy construction.

Findings

Unified algebraic description of Ricci and Calabi flows

Construction of formal solutions using Backlund transformations

Extension to higher-order flows and Kahler manifolds

Abstract

There is a common description of different intrinsic geometric flows in two dimensions using Toda field equations associated to continual Lie algebras that incorporate the deformation variable t into their system. The Ricci flow admits zero curvature formulation in terms of an infinite dimensional algebra with Cartan operator d/dt. Likewise, the Calabi flow arises as Toda field equation associated to a supercontinual algebra with odd Cartan operator d/d \theta - \theta d/dt. Thus, taking the square root of the Cartan operator allows to connect the two distinct classes of geometric deformations of second and fourth order, respectively. The algebra is also used to construct formal solutions of the Calabi flow in terms of free fields by Backlund transformations, as for the Ricci flow. Some applications of the present framework to the general class of Robinson-Trautman metrics that describe spherical gravitational radiation in vacuum in four space-time dimensions are also discussed. Further iteration of the algorithm allows to construct an infinite hierarchy of higher order geometric flows, which are integrable in two dimensions and they admit immediate generalization to Kahler manifolds in all dimensions. These flows provide examples of more general deformations introduced by Calabi that preserve the Kahler class and minimize the quadratic curvature functional for extremal metrics.