In addition to general manifolds with density, the premier example of Gauss space makes an important appearance in Perelman's proof. In summary, manifolds with density provide a convenient technical context for applying diffeomorphisms to focus attention on regions of concern. As Perelman puts it, “The remarkable fact here is that different choices of lead to the same flow, up to a diffeomorphism that is, the choice of is analogous to the choice of gauge.” This means that during the proof one can choose any density for convenience. Remarkably, modulo diffeomorphisms, this generalized Ricci flow is equivalent to the standard Ricci flow. (Actually, in the proof, to facilitate surgery, Perelman moves to a localized version of the density called the length function.) His general monotonicity of energy now provides the requisite local information instead of the usual global information. If one fixes the measure, then the density evolves as a modified backwards heat equation and approaches a delta function at p. To focus attention about a point p of concern, Perelman gives the manifold large density about that point and lets the metric flow by the associated generalized Ricci curvature Ric ψ 18.3( 4). The fundamental difficulty is to obtain some control over the formation of singularities. If you can show that you eventually end up with a round sphere, with perhaps other spheres pinched off along the way, you can conclude that you must have started with a (deformed) sphere. The basic idea of Perelman's proof, due to Richard Hamilton, is to start with any simply connected compact three-manifold and let it shrink at each point in each direction at a rate proportional to its Ricci curvature. In 2003, Grigori Perelman (right) proved it. In 1904, Henri Poincaré (left) proposed his famous conjecture.
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