# Recent Progress on Two Problems in Several Complex Variables

@inproceedings{Ji2007RecentPO, title={Recent Progress on Two Problems in Several Complex Variables}, author={Xiaojun Ji and Wanke Yin}, year={2007} }

We discuss the recent progress on two problems in Several Complex Variables. The first one is on the gap phenomenon for proper holomorphic maps between balls. The second one is on the precise holomorphic structure of Bishop surfaces near a vanishing Bishop invariant.

## 11 Citations

A THEOREM ON HERMITIAN RANK AND MAPPING PROBLEMS

- Mathematics
- 2021

In this paper, we first prove a Huang’s lemma type result. Then we discuss its applications in studying rigidity problems of mappings into indefinite hyperbolic spaces and bounded symmetric domains.…

Upper boundary points of the gap intervals for rational maps between balls

- Mathematics
- 2018

The paper focuses on the study of rational proper holomorphic maps from B to B . We classify these maps when N is the upper boundary point of the gap interval Ik, k ≤ n− 2 and the geometric rank of…

Complexity of holomorphic maps from the complex unit ball to classical domains

- Mathematics
- 2016

We study the complexity of holomorphic isometries and proper maps from the complex unit ball to type IV classical domains. We investigate on degree estimates of holomorphic isometries and holomorphic…

A hyperplane restriction theorem for holomorphic mappings and its application for the gap conjecture

- Mathematics
- 2021

We established a hyperplane restriction theorem for the local holomorphic mappings between projective spaces, which is inspired by the corresponding theorem of Green for OPn(d). Our theorem allows us…

The CR Immersion into a Sphere with the Degenerate CR Gauss Map

- MathematicsThe Journal of Geometric Analysis
- 2019

It is a classical problem in algebraic geometry to characterize the algebraic subvariety by using the Gauss map. In this note, we study the analogous phenomenon in CR geometry. In particular, under…

D’Angelo conjecture in the third gap interval

- MathematicsMathematische Zeitschrift
- 2019

We show that the D’Angelo conjecture holds in the third gap interval. More precisely, we prove that the degree of any rational proper holomorphic map from $${\mathbb {B}}^n$$ B n to $${\mathbb…

On the rank of Hermitian polynomials and the SOS Conjecture

- Mathematics
- 2021

Let z ∈ C and ‖z‖ be its Euclidean norm. Ebenfelt proposed a conjecture regarding the possible ranks of the Hermitian polynomials in z, z̄ of the form A(z, z̄)‖z‖2, known as the SOS Conjecture, where…

Mapping B n into B 3 n − 3

- Mathematics
- 2016

Denote by Rat(Bn,BN ) the collection of all proper holomorphic rational maps from the unit ball Bn ⊂ Cn to the unit ball BN ⊂ CN , and denote by Rat(Hn,HN ) the collection of all proper holomorphic…

On the third gap for proper holomorphic maps between balls

- Mathematics
- 2012

Let $$F$$F be a proper rational map from the complex ball $$\mathbb B ^n$$Bn into $$\mathbb B ^N$$BN with $$n>7$$n>7 and $$3n+1 \le N\le 4n-7$$3n+1≤N≤4n-7. Then $$F$$F is equivalent to a map $$(G, 0,…

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