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The Ricci flow on compact 2-orbifolds with curvature negative somewhere. (English) Zbl 0745.58047
Let \(M\) 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.

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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[1] and , Comparison Theorems in Riemannian Geometry, North-Holland, 1975. · Zbl 0309.53035
[2] The Ricciflow on the 2-sphere, J. Diff. Geom., in press.
[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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