UC Riverside

The structure of an open complete Riemannian manifold (Mn,g) with nonnegative

sectional curvature has been studied extensively and well understood. There are

two classical results due to Gromoll-Meyer [9] and Cheeger-Gromol [4]. Gromoll and Meyer proved that a complete open manifold (Mn,g) with positive sectional curvature is diffeormorphic to Rn. On the other hand, Cheeger and Gromoll proved that a complete open manifold (Mn,g) with nonnegative sectional curvature admits a totally geodesic compact submanifold S such that Mn is diffeomorphic to the normal bundle of S in Mn.

It is natural to imagine that these results and many others can easily be attained in

Ricci curvature case. However, in this case, there are relatively few structural results

except in a lower dimensional case n = 2 where all notions of curvature coincide. In

[17], Shen proved that a complete open Riemannian manifold with nonnegative Ricci

curvature and maximum volume growth is proper (admits an exhaustion function).

Regarding Shen’s result, it was observed by Wong and Zhang [21] that a complete open

K¨ ahler manifold with positive bisectional curvature and maximum volume growth can be embedded as a complex submanifold in a complex Euclidean space of higher dimension. Their observation is a partial result of a weaker version of Yau’s conjecture

which states that a complete open K¨ ahler manifold with positive bisectional curvature

can be embedded as a complex submanifold in a complex Euclidean space of higher

dimension. The original Yau’s conjecture [20] states that: a complete open K¨ ahler

manifold with positive bisectional curvature is biholomorphic to complex Euclidean

space.

Here, we exhibit that a complete open K¨ ahler manifold with positive bisectional

curvature can be embedded as a complex submanifold in a complex Euclidean space of

higher dimension if the volume of a cone of rays from a fixed base point is asymptotic

to the volume of a geodesic ball centered at the same point. The volume growth

condition we consider here is weaker than the maximum volume growth condition.