# The entropy formula for the Ricci flow and its geometric applications

@article{Perelman2002TheEF, title={The entropy formula for the Ricci flow and its geometric applications}, author={Grisha Perelman}, journal={arXiv: Differential Geometry}, year={2002} }

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… Expand

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