Xianghong Gong

Singular Levi-flat real analytic hypersurfaces

We initiate a systematic local study of singular Levi-flat real analytic hypersurfaces, concentrating on the simplest nontrivial case of quadratic singularities. We classify the possible tangent cones to such hypersurfaces and prove the existence and convergence of a rigid normal form in the case of generic (Morse) singularities. We also characterize when such a hypersurface is defined by the vanishing of the real part of a holomorphic function. The main technique is to control the behavior of the homorphic Segre varieties contained in such a hypersurface. Finally, we show that not every such singular hypersurface can be defined by the vanishing of the real part of a holomorphic or meromorphic function, and give a necessary condition for such a hypersurface to be equivalent to an algebraic one.

Normalization of complex-valued planar vector fields which degenerate along a real curve

Abstract Taking as a start point the recent article of Meziani [7], we present several results concerning the normalization of a class of complex vector fields in the plane which degenerate along a real curve. We mainly deal with operators with finite regularity and analyze both the local situation as well as the case of normalization near a circle. Some related questions (e.g., on semi-global solvability and on the normalization of a class of generalized Mizohata operators) are also discussed.

DIRICHLET AND NEUMANN PROBLEMS FOR PLANAR DOMAINS WITH PARAMETER

Let ( ·,�) be smooth, i.e. C 1 , embeddings from onto � , where and � are bounded domains with smooth boundary in the complex plane andvaries in I = (0,1). Suppose that is smooth on × I and f is a smooth function on @ × I. Let u(·,�) be the harmonic functions onwith boundary values f(·,�). We show that u(( z,�),�) is smooth on ×I. Our main result is proved for suitable Holder spaces for the Dirichlet and Neumann problems with parameter. By observing that the regularity of solutions of the two problems with parameter is not local, we show the existence of smooth embeddings ( ·,�) from D, the closure of the unit disc, ontosuch that is smooth on D × I and real analytic at ( √ −1,0) ∈ D × I, but for every family of Riemann mappings R(·,�) fromonto D, the function R(( z,�),�) is not real analytic at ( √ −1,0) ∈ D×I.

Conformal maps and non-reversibility of elliptic area-preserving maps

It has been long observed that area-preserving maps and reversible maps share similar results. This was certainly known to G.D. Birkhoff [5] who showed that these two types of maps have periodic orbits near a general elliptic fixed point. The KAM theory, developed by Kolmogorov-ArnoldMoser for Hamiltonian systems [9], [1] and area preserving maps [15], has also been extended a great deal to reversible systems and maps (see [16], [2], [21]). A natural question is if area-preserving maps and Hamiltonian systems are reversible. In this paper we shall prove

On totally real spheres in complex space

We shall prove that there are totally real and real analytic embeddings of

Fixed points of elliptic reversible transformations with integrals

S^k

The \(\overline{\partial }\)-equation on variable strictly pseudoconvex domains

in

ANTI-HOLOMORPHICALLY REVERSIBLE HOLOMORPHIC MAPS THAT ARE NOT HOLOMORPHICALLY REVERSIBLE

\cc^n

Existence of real analytic surfaces with hyperbolic complex tangent that are formally but not holomorphically equivalent to quadrics

which are not biholomorphically equivalent if

On the convergence of normalizations of real analytic surfaces near hyperbolic complex tangents

k\geq 5