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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.

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
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