Editing Ricci flow (section)

==Li–Yau inequalities==
Making use of a technique pioneered by [[Peter Li (mathematician)|Peter Li]] and [[Shing-Tung Yau]] for parabolic differential equations on Riemannian manifolds, {{harvtxt|Hamilton|1993a}} proved the following "Li–Yau inequality".<ref>{{cite journal |last1=Li |first1=Peter |last2=Yau |first2=Shing-Tung |s2cid=120354778 |title=On the parabolic kernel of the Schrödinger operator |journal=Acta Math. |date=1986 |volume=156 |issue=3–4 |pages=153–201|doi=10.1007/BF02399203 |doi-access=free }}</ref>
* Let <math>M</math> be a smooth manifold, and let <math>g_t</math> be a solution of the Ricci flow with <math>t\in(0,T)</math> such that each <math>g_t</math> is complete with bounded curvature. Furthermore, suppose that each <math>g_t</math> has nonnegative curvature operator. Then, for any curve <math>\gamma:[t_1,t_2]\to M</math> with <math>[t_1,t_2]\subset (0,T)</math>, one has <math display="block"> \frac{d}{dt} \big(R^{g(t)}(\gamma(t))\big)+\frac{R^{g(t)}(\gamma(t))}{t}+\frac{1}{2}\operatorname{Ric}^{g(t)}(\gamma'(t),\gamma'(t))\geq 0.</math>
{{harvtxt|Perelman|2002}} showed the following alternative Li–Yau inequality.
* Let <math>M</math> be a smooth closed <math>n</math>-manifold, and let <math>g_t</math> be a solution of the Ricci flow. Consider the backwards heat equation for <math>n</math>-forms, i.e. <math>\tfrac{\partial}{\partial t}\omega + \Delta^{g(t)}\omega=0</math>; given <math>p\in M</math> and <math>t_0\in(0,T)</math>, consider the particular solution which, upon integration, converges weakly to the Dirac delta measure as <math>t</math> increases to <math>t_0</math>. Then, for any curve <math>\gamma:[t_1,t_2]\to M</math> with <math>[t_1,t_2]\subset (0,T)</math>, one has <math display="block"> \frac{d}{dt} \big(f(\gamma(t),t)\big) + \frac{f\big(\gamma(t),t\big)}{2(t_0-t)} \leq \frac{R^{g(t)}(\gamma(t))+ |\gamma'(t)|_{g(t)}^2}{2}.</math> where <math>\omega=(4\pi(t_0-t))^{-n/2}e^{-f}\text{d}\mu_{g(t)}</math>.
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 {{harvtxt|Chow|Lu|Ni|2006|loc=Chapters 10 and 11}} for details on Hamilton's Li–Yau inequality; the books {{harvtxt|Chow|Chu|Glickenstein|Guenther|2008}} and {{harvtxt|Müller|2006}} contain expositions of both inequalities above.