This is the very first paper to focus on the CR analogue of Yau’s uniformization conjecture in a complete noncompact pseudohermitian (2n+1)-manifold of vanishing torsion (i.e. Sasakian manifold) which is an odd dimensional counterpart of K¨ahler geometry. In this paper, we mainly deal with the problem of the sharp dimension estimate of CR holomorphic functions in a complete noncompact pseudohermitian manifold of vanishing torsion with nonnegative pseudohermitian bisectional curvature.

We extend recent existence results for compact constant mean curvature normal geodesic graphs with boundary in space forms in a unified way to a large class of ambient spaces. That extension is achieved by solving a constant mean curvature existence problem in a general setting presented in two isometrically equivalent versions.

Let ($M$,$ω$) be a connected, symplectic 4-manifold. A$semitoric integrable system$on ($M$,$ω$) essentially consists of a pair of independent, real-valued, smooth functions$J$and$H$on$M$, for which$J$generates a Hamiltonian circle action under which$H$is invariant. In this paper we give a general method to construct, starting from a collection of five ingredients, a symplectic 4-manifold equipped a semitoric integrable system. Then we show that every semitoric integrable system on a symplectic 4-manifold is obtained in this fashion. In conjunction with the uniqueness theorem proved recently by the authors, this gives a classification of semitoric integrable systems on 4-manifolds, in terms of five invariants.

In this paper, we prove the Lipschitz regularity of continuous harmonic maps from an finite dimensional Alexandrov space to a compact smooth Riemannian manifold. This solves a conjecture of F. H. Lin in [36]. The proof extends the argument of Huang-Wang [26].

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