The program of Hamilton is an idea of “plan of attack”, due to Richard Hamilton, certain problems in topology of the varieties, in particular celebrates it Conjecture of Poincaré.

We will try to describe here, without returning in the details, the reasons to be of this program.

Naive idea

Into its article founder of 1982, Three-manifolds with positive Ricci curvature , Richard Hamilton introduces the flood of Ricci . This one is a partial derivative equation bearing on the metric Tenseur of a Variété riemannienne: One starts from metric a $g\_0$, that one makes evolve/move by:

$\backslash \; partial\_t\; g\_t\; =\; -\; 2\; \backslash \; mathrm\; \{Ric\}\; (g\_t),$

where $\backslash \; mathrm\; \{Ric\}$ is the curve of Ricci of the metric one.

It is easy to check that varieties with constant curve, i.e. those provided with a Métrique of Einstein, are solitons or of the generalized fixed points of the flood: the flood of Ricci acts on them only by one dilation.

One can then think (and the first results of Hamilton on the varieties of dimension 3, like on the curves and surfaces confirm this impression) that, just as the equation of heat tends to homogenize a distribution of temperature, the flood of Ricci “will tend” to homogenize the curve of the variety.

To tackle certain problems of topology, Hamilton thus thinks of taking a variety, of providing it with metric a riemannienne, of letting act the flood and of hoping to recover a variety provided with metric with constant curve. For example, if one starts from a simply related variety, and that one thus recovers a simply related variety of strictly positive constant curve, one knows that the variety is not other than the sphere.

Let us moderate this matter: even in the most naive version of its program, Richard Hamilton forever thought of also easily obtaining results of topology. One can extremely well imagine that the flood stops in a finished time, because the curve of the variety explodes, either overall, or locally. The idea would be then to include/understand and classify such “singularities”, and to succeed, using cuttings, to obtain several varieties on which one could start again the flood. The hope is that in fine , one arrives, after a finished number of cuttings, at pieces with constant curve. One would obtain thus, “with the limit”, if all occurs well, of the pieces of varieties which one could know certain topological properties. Our starting variety would thus be obtained in “resticking” these pieces sympathetic nerves, which opens an access road to the famous conjectures of topology, like the Conjecture of Thurston.

Existence in small time

Like any ÉDP, the equation of the flood of Ricci does not check a priori principles of existence and unicity which is comparable with the Théorème of Cauchy-Lipschitz for ÉDO. The first work of Hamilton was to prove that this flood exists in small time.

Let us dare a comparison: it is known that for the equation of heat, there exists a single solution in all times, and that it is infinitely derivable. However, by certain sides (parabolicness), the flood of Ricci resembles the equation of heat. The parabolic equations have a developed general theory, which ensures the existence in small time of solutions. However, the equation of the flood of Ricci is not strictly speaking parabolic: it is only slightly parabolic : the existence and unicity in small time are thus not guaranteed by a general result.

One of the first results of the Hamilton, and by far most fundamental in the study of the flood is thus to prove this result: it reached that point in the article already quoted, by using artillery of the Théorème of inversion of Nash-Moser. However, De Turck arrived at the same result in its article Deforming metrics in the direction off their Ricci tensors of 1983, appeared in the Journal off Differential Geometry , while astutely being reduced to the general theory strictly parabolic equations.

Principles of the maximum

One of the principal analytical tools of the study of the flood of Ricci is the whole of the principles of the maximum. Those make it possible to control certain geometrical quantities (mainly but not only curves) according to their values at the beginning of the flood. More usefully than long a glose, we will state simplest of them: the principle of the scalar maximum.

That is to say $(M,\; G)$ a riemannienne variety and $(g\_t)\; \_\; \{T\}$ a family of metric solutions of the flood of Ricci on the interval $$ and $u:\; M\; \backslash \; times\; \backslash \; to\; \backslash \; mathbb\; checking\; R$ an infinitely derivable function: $\backslash \; partial\_t\; U\; (\backslash \; cdot,\; T)\; +\; \backslash \; triangle\_\; \{g\_t\}\; U\; (\backslash \; cdot,\; T)\; \backslash \; geq\; 0$. Then, for all $t\; \backslash \; in$, $u\; (\backslash \; cdot,\; T)\; \backslash \; geq\; \backslash \; min\_\; \{X\; \backslash \; in\; M\}\; U\; (X,\; 0)$.

There exist more complicated versions of this principle: one can indeed want an assumption less constraining on the equation than checks $u$, or want to rather apply it to tensor than with scalar functions, but the idea is there: with a ÉDP on $u$, one deduces a control in time starting from a control in the beginning.

These results justify that one seeks to determine which equations check the geometrical magnitudes associated with metric, like the curve. Thus of the equation checked by the scalar curve $R$:

$\backslash \; partial\_t\; R\_\; \{g\_t\}\; =\; -\; \backslash \; triangle\_\; \{g\_t\}\; R\_\; \{g\_t\}\; +\; 2\; |\backslash textrm\{Ric\}\_\{g\_t\}|\_\; \{g\_t\}\; ^2,$

one, can deduce that, under the flood of Ricci, the minimum of the scalar curve grows.

A particularly strong version of principle of the maximum, the theorem of pinching of Hamilton and Ivey, valid only in dimension three, affirms that under the flood of Ricci, the sectionnelles curves remain controlled by the scalar curve. This theorem is fundamental in the study of the flood of Ricci, and its absence in higher dimension is one of the causes of the scarcity of the results.

The program today

In three resounding articles ( The entropy formulated for the Ricci flow and its geometric applications, Ricci flow with surgery one three-manifolds and Finite extinction time for the solutions to the unquestionable Ricci flow one three-manifolds ), the Russian mathematician Grigori Perelman presented new ideas to complete the program of Hamilton. It should be noted that Perelman did not subject any of these articles to a mathematical review and that they are available on the Web site of diffusion of prépublications arXiv. In these articles, Perelman claims to classify all the singularities (the $\backslash \; kappa$-solutions) and to make cross icelles to the flood. Perelman affirms that they constitute a proof of the conjecture of Thurston and thus of that of Poincaré.

Having been subjected to no review, these articles have of another raison d'être read only their extraordinary range. Their extreme difficulty and technicality make that their reading is business of years of full-time work for famous mathematicians. Since the international Congress of mathematics of 2006, and even if the ICM explicitly do not quote the conjecture of Poincaré in its presentation of Perelman, the idea that the program of Hamilton is indeed completed is increasingly widespread in the mathematical community.