Editing Ricci flow (section)

==Existence and uniqueness==
Let <math>M</math> be a smooth closed manifold, and let <math>g_0</math> be any smooth Riemannian metric on <math>M</math>. Making use of the [[Nash–Moser theorem|Nash–Moser implicit function theorem]], {{harvtxt|Hamilton|1982}} showed the following existence theorem:
* There exists a positive number <math>T</math> and a Ricci flow <math>g_t</math> parametrized by <math>t\in(0,T)</math> such that <math>g_t</math> converges to <math>g_0</math> in the <math>C^\infty</math> topology as <math>t</math> decreases to 0.
He showed the following uniqueness theorem:
* If <math>\{g_t:t\in(0,T)\}</math> and <math>\{\widetilde{g}_t:t\in(0,\widetilde{T})\}</math> are two Ricci flows as in the above existence theorem, then <math>g_t=\widetilde{g}_t</math> for all <math>t\in(0,\min\{T,\widetilde{T}\}).</math>
The existence theorem provides a one-parameter family of smooth Riemannian metrics. In fact, any such one-parameter family also depends smoothly on the parameter. Precisely, this says that relative to any smooth coordinate chart <math>(U,\phi)</math> on <math>M</math>, the function <math>g_{ij}:U\times(0,T)\to\mathbb{R}</math> is smooth for any <math>i,j=1,\dots,n</math>.

[[Dennis DeTurck]] subsequently gave a proof of the above results which uses the Banach implicit function theorem instead.<ref>{{cite journal |last1=DeTurck |first1=Dennis M. |title=Deforming metrics in the direction of their Ricci tensors |journal=J. Differential Geom. |date=1983 |volume=18 |issue=1 |pages=157–162|doi=10.4310/jdg/1214509286 |doi-access=free }}</ref> His work is essentially a simpler Riemannian version of [[Yvonne Choquet-Bruhat]]'s well-known proof and interpretation of well-posedness for the [[Einstein field equations|Einstein equations]] in Lorentzian geometry.

As a consequence of Hamilton's existence and uniqueness theorem, when given the data <math>(M,g_0)</math>, one may speak unambiguously of ''the'' Ricci flow on <math>M</math> with initial data <math>g_0</math>, and one may select <math>T</math> to take on its maximal possible value, which could be infinite. The principle behind virtually all major applications of Ricci flow, in particular in the proof of the Poincaré conjecture and geometrization conjecture, is that, as <math>t</math> approaches this maximal value, the behavior of the metrics <math>g_t</math> can reveal and reflect deep information about <math>M</math>.