# On the third gap for proper holomorphic maps between balls

@article{Huang2012OnTT, title={On the third gap for proper holomorphic maps between balls}, author={Xiaojun Huang and Shanyu Ji and Wanke Yin}, journal={Mathematische Annalen}, year={2012}, volume={358}, pages={115-142} }

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 $$(G, 0, \dots , 0)$$(G,0,⋯,0) where $$G$$G is a proper holomorphic map from $$\mathbb B ^n$$Bn into $$\mathbb B ^{3n}$$B3n.

## 36 Citations

On local holomorphic maps preserving invariant (p,p)-forms between bounded symmetric domains

- Mathematics
- 2015

Let $D, \Omega_1, ..., \Omega_m$ be irreducible bounded symmetric domains. We study local holomorphic maps from $D$ into $\Omega_1 \times... \Omega_m$ preserving the invariant $(p, p)$-forms induced…

Sum of squares conjecture: the monomial case in $$\mathbb {C}^3$$

- Mathematics
- 2021

The goal of this article is to prove the Sum of Squares Conjecture for real polynomials $$r(z,\bar{z})$$
on $$\mathbb {C}^3$$
with diagonal coefficient matrix. This conjecture describes the…

Holomorphic Deformations of Real-Analytic CR Maps and Analytic Regularity of CR Mappings

- Mathematics
- 2017

Let $$M\subset {\mathbb {C}}^N$$M⊂CN and $$M'\subset {\mathbb {C}}^{N'}$$M′⊂CN′ be real-analytic CR submanifolds, with M minimal. We provide a new sufficient condition, that happens to be also…

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…

Mappings between balls with geometric and degeneracy rank two

- MathematicsScience China Mathematics
- 2018

The paper is devoted to the study of rational proper holomorphic maps from the unit ball $$\mathbb{B}^n$$Bn to the unit ball $$\mathbb{B}^N$$BN. We classify these maps with both the geometric rank…

Proper holomorphic mappings into $\ell$-concave quadric domains in projective space

- Mathematics
- 2013

In this paper, we prove a type of partial rigidity result for proper holomorphic mappings of certain $\ell$-concave domains in projective space into model quadratic $\ell$-concave domains. The main…

Bianalytic maps between free spectrahedra

- Mathematics
- 2016

Linear matrix inequalities (LMIs) $$I_d + \sum _{j=1}^g A_jx_j + \sum _{j=1}^g A_j^*x_j^* \succeq 0$$Id+∑j=1gAjxj+∑j=1gAj∗xj∗⪰0 play a role in many areas of applications. The set of solutions of an…

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…

## References

SHOWING 1-10 OF 80 REFERENCES

Rigidity of CR-immersions into spheres

- Mathematics
- 2002

We consider local CR-immersions of a strictly pseudoconvex real hypersurface $M\subset\bC^{n+1}$, near a point $p\in M$, into the unit sphere $\mathbb S\subset\bC^{n+d+1}$ with $d>0$. Our main result…

A gap rigidity for proper holomorphic maps from $ \B^{n+1}$ to $ \B^{3n-1}$

- Mathematics
- 2006

Let $ \B^{n+1} \subset \C^{n+1}$ be the unit ball in a complex Euclidean space, and let $ \Sigma^n = \partial \B^{n+1} = S^{2n+1}$. Let $ f: \Sigma^n \hook \Sigma^{N}$ be a local CR immersion.If $…

Regularity of CR mappings between algebraic hypersurfaces

- Mathematics
- 1996

We prove that if $M$ and $M'$ are algebraic hypersurfaces in $ C^ N$, i.e. both defined by the vanishing of real polynomials, then any sufficiently smooth CR mapping with Jacobian not identically…

A new gap phenomenon for proper holomorphic mappings from B^n into B^N

- Mathematics
- 2006

In this paper (Math. Res. Lett. 13 (2006). No 4, 509-523), the authors established a pseudo-normal form for proper holomoprhic mappings between balls in complex spaces with degenerate rank. This then…

Maps from the two-ball to the three-ball

- Mathematics
- 1982

In [4], Webster proved the following: Theorem. Let B . = { z e C " : ][7,[]2 3, is a proper holomorphic map which is C a up to the boundary of B,, then f is linear fractional. The purpose of this…

Polynomial and Rational Maps between Balls

- Mathematics
- 2010

Let B be the unit ball in the complex space C. Write Rat(B,B) for the space of proper rational holomorphic maps from B into B and Poly(B,B ) for the set of proper holomorphic polynomial maps from B…

Mapping B n into B 2 n − 1

- Mathematics
- 2001

In this paper, we are concerned with the classification problem of proper holomorphic maps between balls in complex spaces. Write Bn = {z ∈ Cn : |z| < 1} and Prop(Bn, BN ) for the collection of all…

On a linearity problem for proper holomorphic maps between balls in complex spaces of different dimensions

- Mathematics
- 1999

In an important development of several complex variables, Poincaré [26] discovered that any biholomorphic map between two open pieces of the unit sphere in C2 is the restriction of a certain…

A reflection principle with applications to proper holomorphic mappings

- Mathematics
- 1983

j = l holomorphic mappings from B, into B k. We then apply this principle to the problem of characterizing the proper holomorphic mappings from B, into B k. Two such mappings f and g are said to be…

A sharp bound for the degree of proper monomial mappings between balls

- Mathematics
- 2003

The authors prove that a proper monomial holomorphic mapping from the two-ball to the N-ball has degree at most 2N-3, and that this result is sharp. The authors first show that certain…