The topology of complete manifolds with nonnegative Ricci curvature and large volume growth;

For an open complete Riemannian manifold with nonnegative Ricci curvature,the present paper discusses the relation between the topology and the volume growth.

By comparing the volume growth order of the manifold itself to that of its universal covering space, the paper proves that every three-dimensional with nonnegative Ricci curvature and (1+δ)-order volume growth in strict sense must be contractible provided that its universal covering is finite.

And we apply it to two classes of complete noncompact Riemannian manifolds with almost nonnegative Ricci curvature and weak bounded geometry, show that a manifold amon.

A uniformation theorem on complete noncompactn-dimensional(m=2n) Khler manifold with nonnegative and bounded Ricci curvature is studied,if the conditoins as follow are satisfied:① section curvature kr(x0)≥-c/(1+r2);②‖f‖p≤ C0‖▽ f‖q,f∈C∞0(M),1≤q≤n,1/p=1/q-1/m;③ ∫_M Rnic<∞.

Let M be an n(n≥3)-dimensional complete spacelike hypersurface in de Sitter space, S~n+11(1)with constant mean curvature H and constant scalar curvature, it also has nonegative Ricci curvature, then it is isometric to a sphere or an euclidean space or a hyperbolic cylinder.

A property of certain harmonic maps of Ricci curvature which have positive low bound on compact Riemann manifolds,as well as the Eigenvalue estimation problem of harmonic maps are discussed,we get a condition that a harmonic maps is a totally geodesic map.

The paper shows that a complete, noncompact, oriented and strongly stable hypersurface M with constant mean curvature　H in a (n+1)-dimensional complete oriented manifold N~(n+1) with bi-Ricci curvature,being not less than -n~2H~2 along M, admits no nontrivial L~2 harmonic 1-forms.