G. Besson THE GEOMETRIZATION CONJECTURE AFTER R ...

400 G. Bessonu0000 11110000 11110000 11110000 1111distance = r01mthe volume element at distance r from m in the direction of a unit vector u hasan asymptotic expansion given by,d vol = (1 − r2 6 Ricci m(u,u)+o(r 2 ))d vol euclwhere d vol eucl is the Euclidean volume element written in normal coordinates (see [9]).From this it is clear that Ricci is a bilinear form on T m (M) that is an object of the samenature than the Riemannian metric.4.3. The scalar curvatureFinally, the simplest curvature although the weakest is the scalar curvature which is, ateach point, the trace of the bilinear form Ricci with respect to the Euclidean structureg. It is a smooth real valued function on M.5. Hamilton’s Ricci flowThis is an evolution equation on the Riemannian metric g whose expected effect is tomake the curvature look like one in the list of the geometries in 3 dimensions. Thisexpectation is however far too optimistic and not yet proved to be achieved by thistechnique. Nevertheless, this ”flow” turns out to be sufficiently efficient to prove bothPoincaré and Thurston’s conjectures. The inspiration for this beautiful idea is explainedin [10] and [4]. let (M,g 0 ) be a Riemannian manifold, we are looking for a family ofRiemannian metrics depending on a parameter t ∈ R, such that g(0) = g 0 and,dgdt = −2Ricci g(t) .The coefficient 2 is completely irrelevant whereas the minus sign is crucial. This couldbe considered as a differential equation on the space of Riemannian metrics (see [4]), itis however difficult to use this point of view for practical purposes. It is more efficientto look at it in local coordinates in order to understand the structure of this equation([10]). This turns out to be a non-linear heat equation, which is schematically like∂∂t = ∆ g(t) + Q.Here ∆ is the Laplacian associated to the evoluting Riemannian metric g(t). The minussign in the definition of the Ricci flow ensures that this heat equation is not backwardand thus have solutions, at least for small time. The expression encoded in Q is