On the structure of complete Kähler manifolds with nonnegative curvature near infinity
On the structure of complete Kähler manifolds with nonnegative curvature near infinity
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近无穷大非负曲率完全凯勒流形的结构
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发表时间：
1990
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通讯作者：
Peter Li
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作者：
Peter Li
我们想感谢哈罗德·唐纳利在定理3.3的结论指出,流形M未必会分成黎曼的产物N与黎曼曲面S / D o。正确的结论是,对于每个e M,它的总空间全纯纤维形成了一个完整的黎曼面与边界2完全测地线纤维由N黎曼曲面是同胚的半缸1 x R +和非负高斯曲率。纤维N是具有非负截面曲率的紧致K/ihler流形。此外，M的度规是由2 x n的积度规局部给出的，为了澄清这一点，让我们详细说明这个论点。在第590页第3段，我们证明了在每一端，全纯函数h的水平集在h的正则值处必须是完全测地线的。由于完全测地线子流形的极限本身就是一个光滑的完全测地线子流形，我们得出所有的水平集实际上都是m的完全测地线子流形。为了看到这一点，让我们选择一个或正交框架域{el . . . .，以斯拉}的M，其属性为{el . . . .， e2,，-2}与h， e2,,1 = V fl, em= JVf l的水平集相切。由于Busemann函数fl满足IV fl -= 1，且其积分曲线为测地线，可见ve . . . .e2m_ 1 = 0。为了在局部建立分裂，足以证明对合分布T由{e~，…， e2,，-2}是平行的。这等价于证明对于所有向量X tan于M，对于所有1 < i_< 2m2，对于所有2m1 < v < 2m, (Vxe i, ev) = 0。如果对于某些l < j < 2 m2, X = ej，由于h的水平集完全是测地线子流形，这显然是有效的。这个词
We would like to thank Harold Donnelly for pointing out that in the conclusion of Theorem 3.3, the manifold M might not necessarily split into Riemannian product of N with a Riemann surface S/D o. The correct conclusion is that, for each end e of M, it is the total space of a holomorphic fibration over a complete Riemann surface with boundary 2 with totally geodesic fibres given by N. The Riemann surface is homeomorphic to a half cylinder S 1 x R + and has nonnegative Gaussian curvature. The fibre N is a compact K/ihler manifold with nonnegative sectional curvature. Moreover, the metric of M is locally given by the product metric of 2 x N. To clarify this point, let us elaborate on the argument. On page 590, paragraph 3, we have shown that at each end, the level sets of the holomorphic function h must be totally geodesic at the regular values of H. Since the limit of totally geodesic submanifolds is itself a smooth totally geodesic submanifold, we conclude that all the level sets are in fact totally geodesic submanifolds of M. We claim that locally, the metric splits into a product metric. To see this, let us pick an or thonormal frame field {el . . . . , ezra} of M which has the property that {el . . . . , e2,,-2} are tangent to the level sets of h, e2,,1 = V fl, and em= JVf l . Since the Busemann function fl satisfies IV fll -= 1 and its integral curves are geodesics, we see that V e . . . . e2m_ 1=0. To establish the splitting locally, it suffices to show that the involutive distribution T spanned by {e~, ..., e2,,-2} is parallel. This is equivalent to proving that (Vxe i , e v ) = 0 for all vector X tangent to M, for all 1 < i_< 2 m 2, and for all 2 m 1 < v < 2 m. If X = e j for some l < j < 2 m 2 , this is clearly valid due to the fact that the level sets of h are totally geodesic submanifolds. The term