Analogues of Rossi's map and E. Cartan's classification of homogeneous strongly pseudoconvex 3-dimensional hypersurfaces

Effective Cartan-Tanaka connections on C6-smooth strongly pseudoconvex hypersurfaces M3 of C2

In a recent expository article (Notices of the AMS, 58 (2011), no. 1, 20-27), Ezhov, McLaughlin and Schmalz showed how to perform in an effective way Tanaka's prolongation procedure valid generally for filtered structures of constant type when the distribution is equipped with an integrable complex structure, so as to derive the principal curvature invariants and (co)frame(s) associated to strongly pseudoconvex real hypersurfaces M3 in C2, an approach which is alternative and complementary to the `hyperspherical' connection of Elie Cartan, and to the so-called calculi of Fefferman, of Chern-Mo…

Morse inequalities for Fourier components of Kohn-Rossi cohomology of CR manifolds with $S^1$-action

Let $X$ be a compact connected CR manifold of dimension $2n-1, n\geq 2$ with a transversal CR $S^1$-action on $X$. We study the Fourier components of the Kohn-Rossi cohomology with respect to the $S^1$-action. By studying the Szegö kernel of the Fourier components we establish the Morse inequalities on $X$. Using the Morse inequalities we have established on $X$ we prove that there are abundant CR functions on $X$ when $X$ is weakly pseudoconvex and strongly pseudoconvex at a point.

Classification of homogeneous strictly pseudoconvex hypersurfaces in $\mathbb C^3$

Locally homogeneous strictly pseudoconvex hypersurfaces in $\mathbb C^2$ were classified by E.\,Cartan in 1932. In this work, we complete the classification of locally homogeneous strictly pseudoconvex hypersurfaces in $\mathbb C^3$.

Nonvanishing of Cartan CR curvature on boundaries of Grauert tubes around hyperbolic surfaces

We show that the boundaries of thin strongly pseudoconvex Grauert tubes, with respect to the Guillemin-Stenzel Kähler metric canonically associated with the Poincaré metric on closed hyperbolic real-analytic surfaces, has nowhere vanishing Cartan CR-curvature. This result provides a wealth of examples of compact $3$-dimensional Levi nondegenerate CR manifolds having no CR-umbilical point. We provide two proofs utilizing two recent formulas for determining the Cartan CR-curvature of any local $\mathcal{C}^6$-smooth hypersurfaces in $\mathbb{C}^2$. One was obtained in 2012 by the second named au…

Approximation of holomorphic mappings on strongly pseudoconvex domains

Let D be a relatively compact strongly pseudoconvex domain in a Stein manifold, and let Y be a complex manifold. We prove that the set A(D,Y), consisting of all continuous maps from the closure of D to Y which are holomorphic in D, is a complex Banach manifold. When D is the unit disc in C (or any other topologically trivial strongly pseudoconvex domain in a Stein manifold), A(D,Y) is locally modeled on the Banach space A(D,C^n)=A(D)^n with n=dim Y. Analogous results hold for maps which are holomorphic in D and of class C^r up to the boundary for any positive integer r. We also establish the O…

低正则性下严格伪凸超曲面的球面性与解析性 I

Sphericity and Analyticity of a strictly pdeusoconvex hypersurface in low regularity I

本文将严格伪凸超曲面的解析可正则化理论推广至有限光滑性情形，引入正则化 $(0,1)$ 截面工具，解决了 $\mathbb C^2$ 中低正则性下检验超曲面球面性的开放问题，并推导出 $C^6$ 正则性下 CR 曲率消失超曲面的球面性。