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Line 63: − − It has been long understood that Ricci flow on a 3-manifold which satisfies the conditions of the conjecture will result in a three sphere, unless the flow "pinches" the manifold. Perelman was able to show that these "pinches," or singularities, cannot happen.
Poincaré conjecture (view source)
Revision as of 09:24, December 23, 2009
, 09:24, December 23, 2009→Perelman's Proof
The second of these insights, which is the result of Perelman, is whenever the flow pinched the manifold, one could perform surgery on the manifold which would remove the pinch but leave the structure of the manifold unchanged otherwise.
The second of these insights, which is the result of Perelman, is whenever the flow pinched the manifold, one could perform surgery on the manifold which would remove the pinch but leave the structure of the manifold unchanged otherwise.
The effect of the Ricci flow has much in common mathematically with the diffusion of heat in a surface, a concept called [[entropy]]. For example, look at the heat diffusion equation <math>u_t = \Delta u \ </math>. This equation is precisely an expression for the change in the metric of a manifold over time in Ricci flow if we take <math>u = \ln g_{jj} \ </math>, where j is any number less than the dimension of the manifold and no summation is implied on that index, and g is a metric symmetric on the main diagonal (any relevant manifold can have a coordinate system which create such a metric).
The effect of the Ricci flow has much in common mathematically with the diffusion of heat in a surface, a concept called [[entropy]]. For example, look at the heat diffusion equation <math>u_t = \Delta u \ </math>. This equation is precisely an expression for the change in the metric of a manifold over time in Ricci flow if we take <math>u = \ln g_{jj} \ </math>, where j is any number less than the dimension of the manifold and no summation is implied on that index, and g is a metric symmetric on the main diagonal (any relevant manifold can have a coordinate system which create such a metric).