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Extra resources for Algebra and analysis: problems and solutions
Before stating the theorem, we first discuss the case of quadratic curvature decay in relation to eternal solutions. 3). 1) h~ a long time ~olution giJ with initial data g, and the scalar curvature 'R will satisfy tR ~ C for some constant C uniform on spacetime. 1) for K, = 1. Moreover, it is easy to see that g(t) has uniformly bounded nonnegative holomorphic bisectional curvature. 1) for t 2: 1, and that tR ~ C for some uniform constant C. 3) see [43, 39]. 3.  Let M n be a noncompact complex manifold.
Then the universal cover of M is isometrically biholomorphic to en. 3) can be removed. References  Anderson, E. , On the group of holomorphic automorphisms of en, Invent. Math. 110 (1992), no. 2, 371-388. , On the classification of three-dimensional compact Kahler manifolds of nonnegative bisectional curvature, J. Differential Geom. 19 (1984), 283-297.  Barreira, L. and Pesin, Y. , Lyapunov Exponents and Smooth Ergodic Theory, University Lecture Series v. 23, American Mathematical Society, 2001.
Then the K ahler-Ricci flow with initial data 9 has a long time solution g(t) on M x [0,00). It would be nice if one could remove the condition on the injectivity radius. 3. 3: What can we expect of the the curvature decay rate on complete noncompact Kahler manifolds with positive bisectional curvature? There are some results on the volume and curvature of manifolds with nonnegative holomorphic bisectional curvature in . For example, it was proved that if the scalar curvature of a complete noncompact Kahler manifold with nonnegative holomorphic bisectional curvature decays quadratically in the pointwise and average sense, and if the Ricci curvature is positive at some point, then the manifold must have maximum volume growth.
Algebra and analysis: problems and solutions by Lefort G.