Hölder regularity of horocycle foliations
Hölder regularity of horocycle foliations
复制标题
horocycle叶状结构的Hölder规律
DOI：
10.4310/jdg/1214425216
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发表时间：
1999
影响因子：
2.5
通讯作者：
A. Wilkinson
中科院分区：
文献类型：
--
作者：
Marlies Gerber;A. Wilkinson
设M是C∞非正曲流形. M中的一个时球是泛覆盖M中度量球的极限在M上的投影（见§2）。平球面叶理H是单位切丛T 1 M的叶理，其叶由平球面的单位法向量场组成。[1]虽然对负弯曲流形M的平球面叶理的正则性已经有了广泛的研究，但对非正弯曲流形M的正则性知之甚少。最一般的结果是由P. Eberlein得出的：如果M是完全的且非正弯曲的，则半球是C2，这意味着H的各个叶是C1。进一步地，切分布TH连续地依赖于基点v ∈ T1 M（参见[9]）。除了Eberlein定理之外，光滑性结果主要由反例（[2]，[5]）组成;特别是，在一般紧的非正曲M的情况下，人们所能希望的最好结果是TH是霍尔德连续的。本文证明
Let M be a C∞, nonpositively curved manifold. A horosphere in M is the projection to M of a limit of metric spheres in the universal cover M (see §2). A horospherical foliation H is a foliation of the unit tangent bundle T 1M whose leaves consist of unit normal vector fields to horospheres.1 While regularity of horospherical foliations has been studied extensively for negatively curved manifolds M , considerably less is known in the nonpositively curved case. The most general result is due to P. Eberlein: if M is complete and nonpositively curved, then horospheres are C2, which implies that the individual leaves of H are C1. Further, the tangent distribution TH depends continuously the basepoint v ∈ T 1M (see [9]). Beyond Eberlein’s theorem, smoothness results have consisted mainly of counterexamples ([2],[5]); in particular, the best one could hope for in the case of a general compact, nonpositively curved M is for TH to be Holdercontinuous. In this paper we prove