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fixed arxiv front-end link, updated link to personal website

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edited Feb 23, 2022 at 12:58

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This is an open question, as far as I know. Perelman makes a comment in one of his papers to the effect that he would like to achieve some sort of canonical Ricci flow-with-surgery in space-time (see section 13.2 of his first paper his first, 2002 paper). There are several unresolved issues having to do with the formation of the singularities in Ricci flow that make this question difficult. On the other hand, recently Angenent, Knopf, and Caputo have shown that one may do a canonical surgery in the rotationally symmetric case: Minimally invasive surgery for Ricci flow singularities.Minimally invasive surgery for Ricci flow singularities. There are several simplifications in the rotationally symmetric case which makes their approach possible: the Ricci flow is reduced to a (coupled) ODE, so one may apply the maximum principle to show that there are finitely many singularities, and analyze the asymptotics. Since the Ricci flow becomes rotationally symmetric near a singularity in dimension 3 to first order, there is some hope that their approach should work without the rotationally symmetric hypothesis.

This is an open question, as far as I know. Perelman makes a comment in one of his papers to the effect that he would like to achieve some sort of canonical Ricci flow-with-surgery in space-time (see section 13.2 of his first paper). There are several unresolved issues having to do with the formation of the singularities in Ricci flow that make this question difficult. On the other hand, recently Angenent, Knopf, and Caputo have shown that one may do a canonical surgery in the rotationally symmetric case: Minimally invasive surgery for Ricci flow singularities. There are several simplifications in the rotationally symmetric case which makes their approach possible: the Ricci flow is reduced to a (coupled) ODE, so one may apply the maximum principle to show that there are finitely many singularities, and analyze the asymptotics. Since the Ricci flow becomes rotationally symmetric near a singularity in dimension 3 to first order, there is some hope that their approach should work without the rotationally symmetric hypothesis.

This is an open question, as far as I know. Perelman makes a comment in one of his papers to the effect that he would like to achieve some sort of canonical Ricci flow-with-surgery in space-time (see section 13.2 of his first, 2002 paper). There are several unresolved issues having to do with the formation of the singularities in Ricci flow that make this question difficult. On the other hand, recently Angenent, Knopf, and Caputo have shown that one may do a canonical surgery in the rotationally symmetric case: Minimally invasive surgery for Ricci flow singularities. There are several simplifications in the rotationally symmetric case which makes their approach possible: the Ricci flow is reduced to a (coupled) ODE, so one may apply the maximum principle to show that there are finitely many singularities, and analyze the asymptotics. Since the Ricci flow becomes rotationally symmetric near a singularity in dimension 3 to first order, there is some hope that their approach should work without the rotationally symmetric hypothesis.

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created Aug 14, 2010 at 17:59

Ian Agol

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This is an open question, as far as I know. Perelman makes a comment in one of his papers to the effect that he would like to achieve some sort of canonical Ricci flow-with-surgery in space-time (see section 13.2 of his first paper). There are several unresolved issues having to do with the formation of the singularities in Ricci flow that make this question difficult. On the other hand, recently Angenent, Knopf, and Caputo have shown that one may do a canonical surgery in the rotationally symmetric case: Minimally invasive surgery for Ricci flow singularities. There are several simplifications in the rotationally symmetric case which makes their approach possible: the Ricci flow is reduced to a (coupled) ODE, so one may apply the maximum principle to show that there are finitely many singularities, and analyze the asymptotics. Since the Ricci flow becomes rotationally symmetric near a singularity in dimension 3 to first order, there is some hope that their approach should work without the rotationally symmetric hypothesis.