The algebraic structure of geometric flows in two dimensions

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 ∂/∂t. Likewise, the Calabi flow arises as Toda field equation associated to a supercontinual algebra with odd Cartan operator ∂/∂θ − θ∂/∂t. 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 Bäcklund 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 Kähler manifolds in all dimensions. These flows provide examples of more general deformations introduced by Calabi that preserve the Kähler class and minimize the quadratic curvature functional for extremal metrics.

Non-Commutative Ricci and Calabi Flows

“…In this paper we apply this construction to two examples, the two-dimensional Ricci and Calabi flows. Further applications of group-theoretical methods [11] in models which have internal symmetries related to properties of continual Lie algebras, should definitely include implementation of vertex operator constructions [13,21,19], as well as generalization of algebraic constructions [2] of exact special solutions both in commutative and non-commutative cases.…”

“…which is equivalent to (3.7) with ϕ = Φ, when expressing Ψ through Φ, [2]. Therefore we can interprete the two-dimensional Calabi flow equation as a Toda field equation associated with a supercontinual Lie algebra with the same form of commutative relations (5.1), and the mappings…”

Generalized volumetric foliation from inverted viscous flow

“…In the last few decades, the field of geometric flows has seen great progress. As a consequence, these flows have emerged as an essential tool in diverse disciplines such as material science, geometry analysis, topology, quantum field theory and the solution of partial differential equations, among many others [1]. In this work, we propose a geometric flow for closed surfaces, which is based on the three-dimensional Hele-Shaw injection flow [2].…”

“…Our flow has three main properties: (1) If the surface remains a single component during the flow, then it experimentally converges to a sphere; (2) The free surfaces, at each time-step of the flow, form a collection of encapsulating layers, which yield a generalized foliation; (3) The flow is controllable, as the center of innermost spherical layer can be positioned at various points interior to the input surface. Hence, it is possible to influence the speed of the deformation of different parts of the surface, and the progress of the flow.…”

Segmentação de imagens de alta dimensão por meio de algorítmos de detecção de comunidades e super pixels

“…Esta técnica foi proposta por Levinshtein et al (2009) e é baseada na dilatação de sementes mediante fluxos geométricos (Bakas, 2005). A ideia básica é criar e dilatar sementes cujas bordas contenham os super pixels.…”