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On the entropy estimate for the Ricci flow on compact 2-orbifolds. (English) Zbl 0734.53034
Let (M,g) be a compact 2-dimensional Riemannian manifold. Hamilton’s Ricci flow is the solution of the equation $$(\partial /\partial t)g(x,t)=(r-R(x,t))g(x,t),$$ $$x\in M$$, $$t>0$$, where R is the scalar curvature of g and r is the average of R. The entropy -N(t) is defined by $$N(t)=\int_{M_ t}R\cdot \log R dA$$ if R is positive. The author gives a proof of the monotonicity of the entropy. Next, the author defines a modified entropy in the case where g has negative curvature somewhere and then proves the boundedness of the modified entropy.

##### MSC:
 53C20 Global Riemannian geometry, including pinching 58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) 35K55 Nonlinear parabolic equations 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 58J35 Heat and other parabolic equation methods for PDEs on manifolds
##### Keywords:
Ricci flow; scalar curvature; entropy
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