This paper mainly focuses on the CR analogue of the three-circle theorem in a complete noncompact pseudohermitian manifold of vanishing torsion being odd dimensional counterpart of Kaehler geometry. In this paper, we show that the CR three-circle theorem holds if its pseudohermitian sectional curvature is nonnegative. As an application, we confirm the first CR Yau uniformization conjecture and obtain the CR analogue of the sharp dimension estimate for CR holomorphic functions of polynomial growth and its rigidity when the pseudohermitian sectional curvature is nonnegative. This is also the first step toward second and third CR Yauís uniformization conjecture. Moreover, in the course of the proof of the CR three-circle theorem, we derive CR sub-Laplacian comparison theorem. Then Liouville theorem holds for positive pseudoharmonic functions in a complete noncompact pseudohermitian (2n + 1)-manifold of vanishing torsion and nonnegative pseudohermitian Ricci curvature.

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Let F be a foliation in a closed 3-manifold with negatively curved fundamental group and suppose that F is almost trans- verse to a quasigeodesic pseudo-Anosov °ow. We show that the leaves of the foliation in the universal cover extend continuously to the sphere at in¯nity; therefore the limit sets of the leaves are continuous images of the circle. One important corollary is that if F is a Reebless, ¯nite depth foliation in a hyperbolic 3-manifold, then it has the continuous extension property. Such ¯nite depth foliations exist whenever the second Betti number is non zero. The result also applies to other classes of foliations, including a large class of foliations where all leaves are dense, and in¯nitely many examples with one sided branching. One extremely useful tool is a detailed understanding of the topological structure and asymptotic properties of the 1-dimensional foliations in the leaves of e F induced by the stable and unstable foliations of the °ow.

Compactifications of symmetric spaces have been constructed by different methods for various applications. One application is to provide the so-called rational boundary components which can be used to compactify locally symmetric spaces. In this paper, we construct many compactifications of symmetric spaces using a uniform method, which is motivated by the Borel-Serre compact- ification of locally symmetric spaces. Besides unifying compactifications of both symmetric and locally symmetric spaces, this uniform construction allows one to compare and relate easily different compactifications, to extend the group action continuously to boundaries of compactifications, and to clarify the structure of the boundaries.

We consider contracting and expanding curvature flows in $\Ss$. When the flow hypersurfaces are strictly convex we establish a relation between the contracting hypersurfaces and the expanding hypersurfaces which is given by the Gau{\ss} map. The contracting hypersurfaces shrink to a point $x_0$ while the expanding hypersurfaces converge to the equator of the hemisphere $\mc H(-x_0)$. After rescaling, by the same scale factor, the rescaled hypersurfaces converge to the unit spheres with centers $x_0$ \resp $-x_0$ exponentially fast in $C^\un(\Ss[n])$.