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The negative gradient flow for the \(L^2\)-integral of Ricci curvature. (English) Zbl 1054.53080

The author introduces the negative gradient flow for the \(L^2\)-integral of the Ricci curvature and proves the short time existence. In dimension 3, the author also proves that (1) if the maximal time interval \([0, T)\) of existence of solution is infinite, the solution \(g(t)\) converges to a flat metric, and (2) if the interval is finite, then as \(t\rightarrow T\), either the curvature is unbounded or the manifold collapses.

MSC:

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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