This paper continues the previous studies in two papers of Huang-Yin [HY3-4] on the flattening problem of a CR singular point of real codimension two sitting in a submanifold in C n+1 with n + 1 ≥ 3, whose CR points are non-minimal. Partially based on the geometric approach initiated in [HY3] and a formal theory approach used in [HY4], we are able to provide a very general flattening theorem for a non-degenerate CR singular point. As an application, we provide a solution to the local complex Plateau problem and obtain the analyticity of the local hull of holomorphy near a real analytic definite CR singular point in a general setting.

We consider the expansion of co-compact convex hypersurfaces in Minkowski space by functions of their curvature, and prove under quite general conditions that solutions are asymptotic to the self-similar expanding hyperboloid. In particular this implies a convergence result for a class of special solution of the cross-curvature ow of negatively curved Riemannian metrics on three-manifolds. ABSTRACT. We consider the expansion of co-compact convex hypersurfaces in Minkowski space by functions of their curvature, and prove under quite general conditions that solutions are asymptotic to the self-similar expanding hyperboloid. In particular, this implies a convergence result for a class of special solutions of the cross-curvature flow of negatively curved Riemannian metrics on three-manifolds.

Write z = (z 1 , · · · , z n , z n+1 ) for the coordinates of C n+1 and [z] = [z 1 , · · · , z n , z n+1 ] for the homogeneous coordinates of CP n . For a polynomial p(z), we define p(z) := p(z). For a connected projective variety V ⊂ CP n , write I V for the ideal consisting of homogeneous polynomials in z that vanish on V . We define the conjugate variety V * of V to be the projective variety defined byApparently the map z → z defines a diffeomorphism from V to V * . When I V has a basis consisting of polynomials with real coefficients, V * = V . Also if V is irreducible and has a smooth piece parametrized by a neighborhood of the origin of a complex Euclidean space through polynomials with real coefficients, then V * = V .Next for [ξ] ∈ V * , we define the Segre variety Q ξ of V associated with ξ bywhich is a subvariety of codimension one in V . Similarly, for [z] ∈ V , we define the Segre variety Q * z of V * associated with z by Q * z = {[ξ] ∈ V * : n+1 j=1 z j ξ j = 0}. It is clear that [z] ∈ Q ξ if and only if [ξ] ∈ Q * z . The Segre family of V is defined to be the projective variety M :Now, we let (M, ω) be an irreducible Hermitian symmetric space of compact type canonically embedded in a certain minimal projective space CP N , that we will describe in detail later in this section. Then under this embedding, its conjugate space M * is just M itself. Taking ω to

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In this short note we are concerned with the Kähler-Einstein metrics near cone type log canonical singularities. By two different approaches, we construct a complete Kähler-Einstein metric with negative scalar curvature in a neighborhood of the cone over a Calabi-Yau manifold, which provides a local model for the future study of the global Kähler-Einstein metrics on singular varieties. In the first approach, we show that the singularity is uniformized by a complex ball and hence the induced metric from the Bergman metric of the ball is a desired one. In the second approach, we obtain a complete Kähler-Einstein metric with negative curvature by using Calabi Ansatz. At last, we show that these two metrics are indeed the same.