It is conjectured that the manifold with nonnegative Ricci curvature and weaked bounded geometry is of finite topological type, if The paper partially solves this conjecture. In the same time, the paper also discusses...It is conjectured that the manifold with nonnegative Ricci curvature and weaked bounded geometry is of finite topological type, if The paper partially solves this conjecture. In the same time, the paper also discusses the volume growth of a manifold with asymptotically nonnegative Ricci curvature.展开更多
Compact Kǎhler manifolds with semi-positive Ricci curvature have been inves-tigated by various authors. From Peternell＇s work, if M is a compact Kǎhler n-manifold with semi-positive Ricci curvature and finite funda...Compact Kǎhler manifolds with semi-positive Ricci curvature have been inves-tigated by various authors. From Peternell＇s work, if M is a compact Kǎhler n-manifold with semi-positive Ricci curvature and finite fundamental group, then the universal cover has a decomposition M≌X1 χ …x Xm, where Xj is a Calabi-Yau manifold, or a hy-perKǎhler manifold, or Xj satisfies H^0（Xj,Ωp） = 0. The purpose of this paper is to generalize this theorem to almost non-negative Ricci curvature Kǎhler manifolds by us-ing the Gromov-Hansdorff convergence. Let M be a compact complex n-manifold with non-vanishing Euler number. If for any ε〉0, there exists a Kǎhler structure （Jε,gε） on M such that the volume Volgε（M） 〈 V, the sectional curvature ｜K（gε）｜〈 A^2, and the Ricci-tensor Ric（gε）〉-εgε, where V and A are two constants independent of ε. Then the fundamental group of M is finite, and M is diffeomorphic to a complex manifold X such that the universal covering of X has a decomposition, X≌X1 x … x Xs, where Xi is a Calabi-Yau manifold, or a hyperKǎhler manifold, or Xi satisfies H^0（Xi, Ωp）={0}, p 〉 0.展开更多
文摘It is conjectured that the manifold with nonnegative Ricci curvature and weaked bounded geometry is of finite topological type, if The paper partially solves this conjecture. In the same time, the paper also discusses the volume growth of a manifold with asymptotically nonnegative Ricci curvature.
文摘Compact Kǎhler manifolds with semi-positive Ricci curvature have been inves-tigated by various authors. From Peternell＇s work, if M is a compact Kǎhler n-manifold with semi-positive Ricci curvature and finite fundamental group, then the universal cover has a decomposition M≌X1 χ …x Xm, where Xj is a Calabi-Yau manifold, or a hy-perKǎhler manifold, or Xj satisfies H^0（Xj,Ωp） = 0. The purpose of this paper is to generalize this theorem to almost non-negative Ricci curvature Kǎhler manifolds by us-ing the Gromov-Hansdorff convergence. Let M be a compact complex n-manifold with non-vanishing Euler number. If for any ε〉0, there exists a Kǎhler structure （Jε,gε） on M such that the volume Volgε（M） 〈 V, the sectional curvature ｜K（gε）｜〈 A^2, and the Ricci-tensor Ric（gε）〉-εgε, where V and A are two constants independent of ε. Then the fundamental group of M is finite, and M is diffeomorphic to a complex manifold X such that the universal covering of X has a decomposition, X≌X1 x … x Xs, where Xi is a Calabi-Yau manifold, or a hyperKǎhler manifold, or Xi satisfies H^0（Xi, Ωp）={0}, p 〉 0.
