Nevanlinna型问题的一个否定答案和一个具有大量负曲率的抛物曲面
A negative answer to Nevanlinna's type question and a parabolic surface with a lot of negative curvature
摘要：
考虑由Speiser图表示的单连通黎曼曲面。Nevanlinna问，如果类型的表面是由平均过剩的图形：是否平均过剩零意味着表面是抛物线和负平均过剩意味着表面是双曲的。Teichmuller举了一个例子，双曲单连通黎曼曲面的平均过剩是零，反驳了第一个这些影响。我们给出了一个单连通抛物黎曼曲面的例子，从而反驳了另一部分。我们还构造了一个例子，一个完全的，单连通的，抛物面无处正曲率，使积分的曲率在任何磁盘上的一个固定的基点是小于-1/2倍的磁盘的面积，其中1/2> 0是一些常数。
英文摘要：
Consider a simply connected Riemann surface represented by a Speiser graph. Nevanlinna asked if the type of the surface is determined by the mean excess of the graph: whether mean excess zero implies that the surface is parabolic and negative mean excess implies that the surface is hyperbolic. Teichmuller gave an example of a hyperbolic simply connected Riemann surface whose mean excess is zero, disproving the first of these implications. We give an example of a simply connected parabolic Riemann surface with negative mean excess, thus disproving the other part. We also construct an example of a complete, simply connected, parabolic surface with nowhere positive curvature such that the integral of curvature in any disk about a fixed basepoint is less than -epsilon times the area of the disk, where epsilon > 0 is some constant.
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