| In this thesis, we will study Poisson equation, Poincare-Lelong equation and uniqueness of Kahler Ricci flow. One of main goals is to solve Poincare-Lelong equation on a class of Kahler manifolds with nonnegative holomorphic bisectional curvature.; First, we will study sufficient conditions so that the Poisson equation Delta u = f has a solution on the complete noncompact manifolds. By the results of [24], in many cases, a complete noncompact Kahler manifold with nonnegative holomorphic bisectional curvature has the property that the scalar curvature decays at least linearly on the average. Hence it is desirable to solve Deltau = f with k(t) = Bot f≤c 1+t on a complete noncompact Riemannian manifold Mn with Ric ≥ 0. Under the stronger assumption that Ric(x) ≥ alnln10+r x 1+r2x ln10+rx for some a > 67(n + 4)2, we are able to solve Deltau = f with the condition of k(t) as above, which is a generalization of a result in [31]. The solution has some properties which will be used to solve the Poincare-Lelong equation in next part. We will also generalize some results in [23] for the Poisson equation to complete noncompact manifolds which satisfy volume doubling and Poincare inequality. Finally, we will prove that if 0infinity tk(t) < infinity , then the Poisson equation Deltau = f is solvable on some Riemannian manifolds with Ric(x) ≥ -n-1K 1+rx 2 , K > 0. However the estimate of the solution is crude.; In [21] and [23], the Poincare-Lelong equation -166u = rho is solved by first solving the Poisson equation Delta u = tracerho. We will follow the method and use some results of [24] to generalize a result in [31]. Let Mn be a complete non-compact Kahler manifold of complex dimension n with nonnegative holomorphic bisectional curvature, bounded scalar curvature and Ric(x) ≥ alnln10+r x 1+r2x ln10+rx for some a > 268(n + 2)2 , we will prove that -166u = Ric has a solution if all group homomorphisms from pi1 (M) to R are trivial, and the universal covering space M&d15; of M with the covering metric has no compact factors.; In the last part of this thesis, we will study uniqueness of Kahler Ricci flows on complete noncompact Kahler manifolds with bounded Riemannian curvature tensor. In [11, 12], Hamilton proved the short time existence and its uniqueness for Ricci flows on any compact manifold. Let (M,g ij¯ (x)) be a complete noncompact Kahler manifold with bounded Riemannian curvature tensor, W.-X. Shi [28, 29] proved the short time existence for the Kahler Ricci flow dgijx,t dt = -Rij¯ (x,t ) with gij¯ (x,0) = gij¯ (x), where Rij¯ (x,t) is the Ricci tensor of gij¯ (x,t). If in addition, the Ricci tensor with respect to gij¯ ( x) has a potential, we will show that two solutions g ij¯ (x,t), g˜ ij¯ (x,t) on M x [0, T] to the Kahler Ricci flow with the same initial value gij¯ (x) are equal, provided that they are Kahler metrics on M and there is a constant C such that C -1gij¯ (x) ≤ gij¯ (x,t), g ij¯ (x,t) ≤ Cg ij¯ (x) for all t ∈ [0, T]. Finally, we will also study the convergence of some Kahler Ricci flows on Cn with the initial metrics constructed by Wu and Zheng [32]. |
