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In this paper, we prove that for any Kähler metrics ω and χ on M, there exists a Kähler metric ω_φ = ω_0 + √-1 ∂ \bar∂ φ > 0 satisfying the J-equation tr_{ω_φ} χ = c if and only if (M, [ω0], [χ]) is uniformly J-stable. As a corollary, we find a sufficient condition for the existence of constant scalar curvature Kähler metrics with c_1 < 0. Using the same method, we also prove a similar result for the supercritical deformed Hermitian–Yang–Mills equation.

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.

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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