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The Ricci flow on $$2$$-orbifolds with positive curvature. (English) Zbl 0735.53030
If a group $$G$$ acts properly discontinuously on a smooth manifold M, then the quotient space is a smooth orbifold. If the universal cover of a smooth orbifold is a manifold, then it is called a good orbifold, otherwise, it is called bad. In this paper, after some preparatory lemmas and nice illustrations, the author proves the following theorem: any metric with positive curvature on a bad orbifold asymptotically approaches a Ricci soliton at time infinity under the Ricci flow, where a soliton is a solution which moves only by diffeomorphism. This theorem provides the first known example where a non-Kähler Einstein orbifold converges to a nontrivial Ricci soliton and a way to get a canonical metric on a bad orbifold.

##### MSC:
 53C20 Global Riemannian geometry, including pinching
##### Keywords:
smooth orbifold; bad orbifold; Ricci soliton; Ricci flow
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