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Function Theory on Open Kahler Manifolds

  • Author(s): Ogaja, James W
  • Advisor(s): Wong, Bun
  • et al.
Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International Public License
Abstract

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.

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