Hausdorff Dimension of Limit Sets for Spherical CR Manifolds

✍ Scribed by Zhongyuan Li

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 667 KB
• Volume: 139
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1996.0077

✦ Synopsis

Let M 2n+1 (n 1) be a compact, spherical CR manifold. Suppose M 2n+1 is its universal cover and 8 : M 2n+1 Ä S 2n+1 is on injective CR developing map, where S 2n+1 is the standard unit sphere in the complex (n+1)-space C n+1 , then M 2n+1 is of the quotient form 0Â1, where 0 is a simply connected open set in S 2n+1 , and 1 is a complex Klein group acting on 0 properly discontinuously. In this paper, we show that if the CR Yamabe invariant of M 2n+1 is positive, then the Carnot Hausdorff dimension of the limit set of 1 is bounded above by n } s(M 2n+1 ), where s(M 2n+1 ) 1 and is a CR invariant. The method that we adopt is analysis of the CR invariant Laplacian. We also explain the geometric origin of this question.