This paper studies local geometry of hypersurfaces of finite multitype. Catlin’s definition of multitype is applied to a general smooth hypersurface in C. We prove biholomorphic equivalence of models in dimension three and describe all biholomorphisms between such models. A finite constructive algorithm for computing multitype is described. Analogous results for decoupled hypersurfaces are given.

This paper considers the number of discrete fixed points that an automorphism (biholomorphic self-map) of a complex domain or manifold can have. Our results give a geometric formulation to the problem and its solution; they generalize classical theorems in one complex

Let H and F be two Hénon maps with biholomorphically equivalent escaping sets, then there exist affine automorphisms $$A_1$$ $$A_2$$ in $${\mathbb {C}}^2$$ such that $$\begin{aligned} F=A_1\circ \circ A_2 \end{aligned}$$ .

In this article we study the limiting behavior of the KählerRicci flow on complete non-compact Kähler manifolds. We provide sufficient conditions under which a complete non-compact gradient KählerRicci soliton is biholomorphic to C. We also discuss the uniformization conjecture by Yau [15] for complete non-compact Kähler manifolds with positive holomorphic bisectional curvature.

In this paper we study the function algebra generated by z and g on a small closed disk centered at the origin of the complex plane. We prove, using a biholomorphic change of coordinates and already developed techniques in this area, that for a large class of functions g this algebra consists of all continuous functions on the disk.

where a, b are real-valued real analytic functions defined near the origin. We always assume that 0 ∈ C, and that all changes of coordinates are local and fix the origin. We say that two parameterized curves C and C̃ are equivalent if there exist a biholomorphic map h and a real analytic diffeomorphism φ so that h(C(t)) = C̃(φ(t)). We also say that C, C̃ are formally equivalent, if h is a formal b...

Let $X$ be a (smooth) compact complex surface. We show that the torsion subgroup of biholomorphic automorphism group $\operatorname{Aut}(X)$ is virtually nilpotent. Moreover, we study Tits alternative and virtual derived length solvable subgroups $\operatorname{Aut}(X)$.

In this paper, we will be concerned with the local biholomorphic properties of a real n-manifold M in C. At a generic point, such a manifold basically has the nature of the standard R in C. Near a complex tangent, however, the consideration can be much more complicated and the manifold may acquire a nontrivial local hull of holomorphy and many other biholomorphic invariants. The study of such a...