Towards physically motivated proofs of the Poincaré and geometrization conjectures

Although the Poincaré 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 [D. Friedan, Nonlinear models in 2+ε2+ε dimensions, Ann. Phys. 163 (1985) 318–419]. In his work the renormalization group flow for a nonlinear sigma model in 2+ε2+ε dimensions was obtained and studied. For ε=0ε=0, by approximating the ββ-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 two-dimensional conformal field theories (CFT’s) to three 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 rather 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 Poincaré and geometrization conjectures.