Ricci Flow and the Poincaré Conjecture

Entropy functionals and thermodynamics of relativistic geometric flows, stationary quasi-periodic Ricci solitons, and gravity

“…Such s-splitting will allow to decouple and integrate in general off-diagonal form nonassociative geometric and physically important systems of nonlinear PDEs. In a local coordinate basis (see conventions from footnotes 7, 8, and 9), a nonlinear s-connection defined by a Whitney sum ⊕ as in (10), for instance, is characterized by coefficients s N = { N isas ( u)}, for u = (x, p) = u = ( 1 x, 2 y, 3 p, 4 p). Such coefficients allow us to construct N-elongated bases (N-/ s-adapted bases) as linear N-operators:…”

“…describing the flow evolution on M of a family of nonassociative metrics g ⋆ (τ ) parameterized by a positive parameter 0 ≤ τ ≤ τ 0 . In these formulas, dots are used for additional terms which can be defined/computed for respective star product R-flux deformations of a corresponding variational calculus [1,9,10,11,7], see references therein and details in section 2.3. 5 The coefficients of geometric objects, effective and matter field sources, and respective physically important geometric evolution/ dynamical equations in such theories can be decomposed into real and complex terms and parametric forms using decompositions on , the Planck constant, and κ := ℓ 3 s /6 , the string constant, were ℓ s is a length parameter [39].…”

“…It should be emphasized that a type of f (τ ) may have different implications and interpretations in topology and/or geometric analysis, and differential geometry theories. 10 Using a (3+1) splitting on 4-d, Lorentz manifolds, the functionals (9) can be generalized in relativistic form, which allows us to prove (using standard variational procedures, or abstract geometric methods) respective generalizations of the R. Hamilton equations [4] and elaborate on relativistic geometric thermodynamic models. This does not results in a formulation and proof of a relativistic version of the Poincaré-Thorston conjecture.…”

“…In principle, those constructions allow to elaborated on nonassociative versions of G. Perelmans functionals (9). Applying rigorous geometric methods and respective variational procedures generalizing the geometric analysis formalism from [1,9,10,11,7], we can derive respective nonassociative geometric flow equaitons. Such star deformed R. Hamilton equations and, in particular nonassociative Ricci solitons and/or nonassociative vacuum Einstein equations consists very sophisticate systems of nonlinear partial differential equations, PDEs.…”

Uniqueness and stability of singular Ricci flows in higher dimensions

“…For the Ricci flow, a surgery process has been first proposed in pioneering work by Hamilton [Ham95,Ham97]. In his spectacular papers [Per02,Per03], Perelman successfully constructed a flow with surgery in three dimensions and used it to prove the Poincare and geometrization conjecture (see also [KL08,MT07,CZ06] for detailed expositions). This construction has been later generalized to other settings, including orbifolds [DL09,KL14],…”