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Deforming Riemannian metrics on complex projective spaces. (English) Zbl 0642.53074

Geometry and partial differential equations, 2nd Miniconf., Canberra/Aust. 1986, Proc. Cent. Math. Anal. Aust. Natl. Univ. 12, 86-95 (1987).
[For the entire collection see Zbl 0626.00019.]
R. S. Hamilton [J. Differ. Geom. 17, 255-306 (1982; Zbl 0504.53034), and “Four-manifolds with positive curvature operator” (in preparation)] and G. Huisken [J. Differ. Geom. 21, 47-62 (1985; Zbl 0606.53026)] have shown a condition for a metric g of a compact Riemannian manifold M to be deformable to a space form, letting \(g_{ij}\) evolve according to the equation \[ \frac{\partial g_{ij}}{\partial t}=(2/n)rg_{ij}-2R_{ij}, \] where \(r=\int_{M}Rd\mu /\int_{M}d\mu\) is the average of the scalar curvature. The aim of this paper is to obtain a condition under which a metric will evolve to a multiple of the Fubini-Study metric on CP \((n=2m)\) according to a system of the similar equations as above. The result, however, is highly complicated, and a short cut for the solution is shown as follows: Let there be given constants \(\delta <1\) and \(\epsilon_ 1\), \(\epsilon_ 2\), \(\epsilon_ 3\) on an almost Hermitian manifold M. If M is \(\delta\)-pinched and its curvature and the almost complex structure J satisfy \(| \nabla R_ m| <\epsilon_ 1\), \(| \nabla J| <\epsilon_ 2\), \(| \nabla \nabla J| <\epsilon_ 3\), then the metric of M can well be evolved to that of a Kaehler manifold of constant positive holomorphic curvature.
Reviewer: T.Okubo

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds