equação Graceli relativista dimensional tensorial  quântica de campos G* = =   RGG[] G [.] [   ] = {[ G* = ] / { [] [.]    , { [] [ ω  , , ] / c }}.

Normalized Ricci flow[edit source]

Suppose that M is a compact smooth manifold, and let g_t be a Ricci flow for t in the interval (a, b). Define Ψ:(a, b) → (0, ∞) so that each of the Riemannian metrics Ψ(t)g_t has volume 1; this is possible since M is compact. (More generally, it would be possible if each Riemannian metric g_t had finite volume.) Then define F:(a, b) → (0, ∞) to be the antiderivative of Ψ which vanishes at a. Since Ψ is positive-valued, F is a bijection onto its image (0, S). Now the Riemannian metrics G_s  =  Ψ(F ⁻¹(s))g_F ⁻¹(s), defined for parameters s ∈ (0, S), satisfy

$${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }�}{�}_{�}=-2{\mathrm{Ric}}^{{�}_{�}}+\frac{2}{�}\frac{{\int }_{�}{�}^{{�}_{�}}�{�}_{{�}_{�}}}{{\int }_{�}�{�}_{{�}_{�}}}{�}_{�}.}$$

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equação Graceli relativista dimensional tensorial  quântica de campos G* = =   RGG[] G [.] [   ] = {[ G* = ] / { [] [.]    , { [] [ ω  , , ] / c }}.

Here R denotes scalar curvature. This is called the normalized Ricci flow equation. Thus, with an explicitly defined change of scale Ψ and a reparametrization of the parameter values, a Ricci flow can be converted into a normalized Ricci flow. The converse also holds, by reversing the above calculations.

The primary reason for considering the normalized Ricci flow is that it allows a convenient statement of the major convergence theorems for Ricci flow. However, it is not essential to do so, and for virtually all purposes it suffices to consider Ricci flow in its standard form. Moreover, the normalized Ricci flow is not generally meaningful on noncompact manifolds.

Li–Yau inequalities[edit source]

Making use of a technique pioneered by Peter Li and Shing-Tung Yau for parabolic differential equations on Riemannian manifolds, Hamilton (1993a) proved the following "Li–Yau inequality."^[5]

• Let ${\displaystyle �}$ be a smooth manifold, and let ${\displaystyle {�}_{�}}$ be a solution of the Ricci flow with ${\displaystyle �\in (0,�)}$ such that each ${\displaystyle {�}_{�}}$ is complete with bounded curvature. Furthermore, suppose that each ${\displaystyle {�}_{�}}$ has nonnegative curvature operator. Then, for any curve ${\displaystyle �:[{�}_1,{�}_2]\to �}$ with ${\displaystyle [{�}_1,{�}_2]\subset (0,�)}$, one has
  $${\displaystyle \frac{�}{��}({�}^{�(�)}(�(�)))+\frac{{�}^{�(�)}(�(�))}{�}+\frac{1}{2}{\mathrm{Ric}}^{�(�)}({�}^{\prime }(�),{�}^{\prime }(�))\ge 0.}$$

  
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equação Graceli relativista dimensional tensorial  quântica de campos G* = =   RGG[] G [.] [   ] = {[ G* = ] / { [] [.]    , { [] [ ω  , , ] / c }}.

Perelman (2002) showed the following alternative Li–Yau inequality.

• Let ${\displaystyle �}$ be a smooth closed ${\displaystyle �}$-manifold, and let ${\displaystyle {�}_{�}}$ be a solution of the Ricci flow. Consider the backwards heat equation for ${\displaystyle �}$-forms, i.e. ${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }�}�+{\mathrm{\Delta }}^{�(�)}�=0}$; given ${\displaystyle �\in �}$ and ${\displaystyle {�}_0\in (0,�)}$, consider the particular solution which, upon integration, converges weakly to the Dirac delta measure as ${\displaystyle �}$ increases to ${\displaystyle {�}_0}$. Then, for any curve ${\displaystyle �:[{�}_1,{�}_2]\to �}$ with ${\displaystyle [{�}_1,{�}_2]\subset (0,�)}$, one has
  $${\displaystyle \frac{�}{��}(�(�(�),�))+\frac{�(�(�),�)}{2({�}_0-�)}\le \frac{{�}^{�(�)}(�(�))+|{�}^{\prime }(�){|}_{�(�)}^2}{2}.}$$

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  equação Graceli relativista dimensional tensorial  quântica de campos G* = =   RGG[] G [.] [   ] = {[ G* = ] / { [] [.]    , { [] [ ω  , , ] / c }}.

  where ${\displaystyle �=(4�({�}_0-�){)}^{-�/2}{�}^{-�}\text{d}{�}_{�(�)}}$./

equação Graceli relativista dimensional tensorial  quântica de campos G* = =   RGG[] G [.] [   ] = {[ G* = ] / { [] [.]    , { [] [ ω  , , ] / c }}.

Both of these remarkable inequalities are of profound importance for the proof of the Poincaré conjecture and geometrization conjecture. The terms on the right hand side of Perelman's Li–Yau inequality motivates the definition of his "reduced length" functional, the analysis of which leads to his "noncollapsing theorem." The noncollapsing theorem allows application of Hamilton's compactness theorem (Hamilton 1995) to construct "singularity models," which are Ricci flows on new three-dimensional manifolds. Owing to the Hamilton–Ivey estimate, these new Ricci flows have nonnegative curvature. Hamilton's Li–Yau inequality can then be applied to see that the scalar curvature is, at each point, a nondecreasing (nonnegative) function of time. This is a powerful result that allows many further arguments to go through. In the end, Perelman shows that any of his singularity models is asymptotically like a complete gradient shrinking Ricci soliton, which are completely classified; see the previous section.

See Chow, Lu & Ni (2006, Chapters 10 and 11) for details on Hamilton's Li–Yau inequality; the books Chow et al. (2008) and Müller (2006) contain expositions of both inequalities above.