Towards physically motivated proofs of the Poincare' and geometrization conjectures

TL;DR

This paper explores physically motivated proofs of the Poincare' and geometrization conjectures by connecting Ricci flows to physical models like conformal field theories, aiming to provide more natural justifications for these mathematical results.

Contribution

It develops a new formalism extending 2D conformal field theory results to higher dimensions, offering independent physical justification for Ricci flows in nature.

Findings

Ricci flows can be realized through critical dynamics of higher-dimensional CFTs

A new formalism links Yamabe and Ricci flows to physical phenomena

Steps toward physically motivated proofs of geometric conjectures

Abstract

Although the Poincare' and the geometrization conjectures were recently proved by Perelman, the proof relies heavily on properties of the Ricci flow previously investigated in great detail by Hamilton. Physical realization of such a flow can be found, for instance, in the work by Friedan (Ann.Phys.163(1985)318-419). In his work the renormalization group flow for a nonlinear sigma model in 2+e dimensions was obtained and studied. For e=0, by approximating the beta function for such a flow by the lowest order terms in the sigma model coupling constant, the equations for Ricci flow are obtained. In view of such an approximation, the existence of this type of flow in nature is questionable. In this work, we find totally independent justification for the existence of Ricci flows in nature. This is achieved by developing a new formalism extending the results of 2d CFT to 3 and higher dimensions. Equations describing critical dynamics of these CFT's are examples of the Yamabe and Ricci flows realizable in nature. Although in the original works by Perelman some physically motivated arguments can be found, their role in his proof remain either nonexistent or obscure. In this paper, steps are made toward making these arguments more explicit thus creating an opportunity for developing alternative, more physically motivated, proofs of the Poincare' and geometrization conjectures.