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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
##### Keywords:
Ricci flow; area estimates
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##### References:
  and , Comparison Theorems in Riemannian Geometry, North-Holland, 1975. · Zbl 0309.53035  The Ricciflow on the 2-sphere, J. Diff. Geom., in press.  The Ricci flow on surfaces, in Mathematics and General Relativity, Contemporary Mathematics, AMS 71, 1988, pp. 237–262. · doi:10.1090/conm/071/954419  Scott, Bull. London Math. Soc. 15 pp 401– (1983)  The Geometry and Topology of 3-Manifolds, Princeton Univ. Lecture Notes, 1979.  Wu, J. Diff. Geom.
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