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arXiv:math/0211159math.DG

The entropy formula for the Ricci flow and its geometric applications

Grisha Perelman

Abstract

We present a monotonic expression for the Ricci flow, valid in all dimensions and without curvature assumptions. It is interpreted as an entropy for a certain canonical ensemble. Several geometric applications are given. In particular, (1) Ricci flow, considered on the space of riemannian metrics modulo diffeomorphism and scaling, has no nontrivial periodic orbits (that is, other than fixed points); (2) In a region, where singularity is forming in finite time, the injectivity radius is controlled by the curvature; (3) Ricci flow can not quickly turn an almost euclidean region into a very curved one, no matter what happens far away. We also verify several assertions related to Richard Hamilton's program for the proof of Thurston geometrization conjecture for closed three-manifolds, and give a sketch of an eclectic proof of this conjecture, making use of earlier results on collapsing with local lower curvature bound.

arXiv comments
39 pages
Journal reference
DOI
Subjects
math.DGDifferential Geometry

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Linked by 5 identifiable commenters in 5 comments across 5 threads, from May 2010 to Mar 2026. Found in 1 HN submission from Sep 2025 to Sep 2025. Open submission discussions.

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Discussed in comments5 threads

  1. ArXiv declares independence from Cornell 1 comment
  2. Beyond Einstein: Physicists find surprising connections in the cosmos 1 comment

    • The starting point of Perelman's proof is the observation that (a modified version of) the Ricci flow is a gradient flow. The connection to 2d non-linear sigma models is briefly discussed in section 2 of the introduction in https://arxiv.org/pdf/math/0211159.pdf and section…

      orbifoldView on HNParent context

  3. Do I really have to cite an arXiv paper? 1 comment
  4. ArXiv Hits 1M Submissions 1 comment

    • One example of this happening: Perlman's proof of the Poincare conjecture was posted on the arXiv first (http://arxiv.org/abs/math/0211159). In fact, I don't think it was ever formally published.

      evanpwView on HNParent context

  5. Ask HN: Publish in peer-review journal without affiliation? 1 comment

    • Here is a notable exception of a non-peer-reviewed paper which did have "some" impact: http://arxiv.org/abs/math.DG/0211159 As a place to stake one's claim(sure beats making anagrams as in 17th century) and disseminate knowledge arxiv seems like a good place to be.

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