The Ricci flow on compact 2-orbifolds with curvature negative somewhere.

*(English)*Zbl 0745.58047Let \(M\) be a compact 2-dimensional orbifold with positive Euler characteristic. Hamilton’s area preserving Ricci flow is given by the equation
\[
{\partial\over \partial t}g(x,t)=(r-R(x,t))g(x,t),\qquad x\in M,\qquad t > 0,
\]
where \(g\) is the metric, \(R\) is the scalar curvature of \(g\), \(r\) is the average curvature of \(R\). A soliton solution \(\{g_ t\}\) of Hamilton’s Ricci flow is defined as \(g_ t=\varphi^*_ t(g_ 0)\), where \(\{\varphi_ t\}\) is some diffeomorphism of \(M\).

The following theorem is proved: Theorem. If \(g\) is any metric on a compact 2-orbifold with the positive Euler characteristic, then under Hamilton’s Ricci flow, \(g\) approaches asymptotically a soliton solution.

This theorem, together with the earlier result by R. S. Hamilton [Contemp. Math. 71, 237-262 (1988; Zbl 0663.53031)] gives a complete solution to

Hamilton’s conjecture. If \((M,g)\) is a compact 2-dimensional Riemannian orbifold, then under the Ricci flow, \(g\) approaches asymptotically a soliton solution.

The following theorem is proved: Theorem. If \(g\) is any metric on a compact 2-orbifold with the positive Euler characteristic, then under Hamilton’s Ricci flow, \(g\) approaches asymptotically a soliton solution.

This theorem, together with the earlier result by R. S. Hamilton [Contemp. Math. 71, 237-262 (1988; Zbl 0663.53031)] gives a complete solution to

Hamilton’s conjecture. If \((M,g)\) is a compact 2-dimensional Riemannian orbifold, then under the Ricci flow, \(g\) approaches asymptotically a soliton solution.

Reviewer: A.Bobenko (St.Petersburg)

##### MSC:

58J60 | Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) |

37C10 | Dynamics induced by flows and semiflows |

53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |

35Q51 | Soliton equations |

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\textit{B. Chow} and \textit{L.-F. Wu}, Commun. Pure Appl. Math. 44, No. 3, 275--286 (1991; Zbl 0745.58047)

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##### References:

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[3] | The Ricci flow on surfaces, in Mathematics and General Relativity, Contemporary Mathematics, AMS 71, 1988, pp. 237–262. · doi:10.1090/conm/071/954419 |

[4] | Scott, Bull. London Math. Soc. 15 pp 401– (1983) |

[5] | The Geometry and Topology of 3-Manifolds, Princeton Univ. Lecture Notes, 1979. |

[6] | Wu, J. Diff. Geom. |

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