Proper maps of ball complements & differences and rational sphere maps

Studying proper holomorphic maps between domains is a common problem in several complex variables. In general, such maps do not exist unless we choose specific domains. If the domain has many symmetries, such as the unit ball $\mathbb{B}_{n}\subset\mathbb{C}^{n}$, many maps exist. Alexander [alexander-1977-proper] proved that every proper holomorphic map $f\colon\mathbb{B}_{n}\to\mathbb{B}_{n}$, $n>1$, is an automorphism, and hence rational. On the other hand, Dor [dor-1990-proper] showed that there exist proper holomorphic maps from $\mathbb{B}_{n}$ to $\mathbb{B}_{n+1}$ that are continuous up to the boundary and are not rational. Forstnerič [forstneric-1989-extending] proved that if for some neighborhood $U$ of a point $p$ on the sphere, a sufficiently smooth $f\colon U\cap\overline{\mathbb{B}_{n}}\to\mathbb{C}^{N}$ is holomorphic on $U\cap\mathbb{B}_{n}$ and $f(U\cap S^{2n-1})\subset S^{2N-1}$, then $f$ is rational and extends to a proper map of $\mathbb{B}_{n}$ to $\mathbb{B}_{N}$. Moreover, the degree of $f$ is bounded by a function of $n$ and $N$ alone. We will make extensive use of Forstnerič’s result in this work. For simplicity, by a proper map of balls, we will always mean a proper holomorphic map $f\colon\mathbb{B}_{n}\to\mathbb{B}_{N}$. Rudin [rudin-1984-homogeneous] proved that every homogeneous polynomial proper map of balls is unitarily equivalent to the symmetrized $m$-fold tensor product of the identity map, while D’Angelo [dangelo-1988-polynomial] found a much simpler proof of Rudin’s result and moreover gave complete procedure for constructing all polynomial proper maps of balls. A key idea in D’Angelo’s work, and one that we will make use of, is that given two vector valued polynomials $p\colon\mathbb{C}^{n}\to\mathbb{C}^{m}$ and $q\colon\mathbb{C}^{n}\to\mathbb{C}^{k}$ such that $\mathopen{}\mathclose{{}\left\lVert p(z)}\right\rVert^{2}=\mathopen{}\mathclose{{}\left\lVert q(z)}\right\rVert^{2}$ for all $z$, there is a unitary map $U$ such that $U(p\oplus 0)=q\oplus 0$. By $\oplus 0$ we mean we add zero components if they are needed to match the target dimensions. For more information on this subject, see [faran-1982-maps, forstneric-1993-proper, huang-1999-linearity, hamada-2005-rational, huang-2006-new, huang-2014-third, lebl-2024-exhaustion] and the references therein. In particular, the books by D’Angelo [dangelo-1993-several, dangelo-2019-hermitian, dangelo-2021-rational] are relevant.

We change the point of view slightly and consider the complements of balls and the differences of balls, that is, without loss of generality we study maps between the sets of the form $\mathbb{C}^{n}\setminus\overline{\mathbb{B}_{n}}$ and $\mathbb{B}_{n}\setminus\overline{B_{r}(c)}$, where $B_{r}(c)$ denotes the ball of radius $r$ centered at $c$. As before, we say proper map of ball complements to mean a proper holomorphic map $f\colon\mathbb{C}^{n}\setminus\overline{\mathbb{B}_{n}}\to\mathbb{C}^{N}\setminus\overline{\mathbb{B}_{N}}$. Similarly, by proper map of ball differences we mean a proper holomorphic map $f\colon\mathbb{B}_{n}\setminus\overline{B_{r}(c)}\to\mathbb{B}_{N}\setminus\overline{B_{R}(C)}$. We remark that proper maps to domains that are complements of balls have been studied previously; see [forstnerivc-2014-oka]. However, we wish the domain of the map to also be a ball complement. First, we characterize all proper maps of ball complements.

The condition on the norm is clearly satisfied for proper maps of complements, but it is not clear if it is automatically true for polynomial proper maps of balls. We prove that the condition is satisfied if the polynomial takes the origin to the origin, or more trivially, if the top degree terms of $p$ do not vanish on the sphere.

That the proper map of ball complements is polynomial follows from a combination of the Hartogs phenomenon and Forstnerič’s theorem mentioned above. The key is then to show that polynomial maps of balls are precisely those that also take the outside to the outside. The theorem does not hold if $n=1$: $\mathbb{C}\setminus\overline{\mathbb{D}}$ is biholomorphic to the punctured disc $\mathbb{D}^{*}$, which properly, and certainly not rationally, embeds into $\mathbb{C}^{N}$ via the classical theorem of Remmert–Bishop–Narasimhan (see [forstneric-2017-stein-manifolds]*Theorem 2.4.1 and also Alexander [alexander-1977-punctured-disc] for an explicit embedding into $\mathbb{C}^{2}$). Such a map certainly avoids some small closed ball. A more complicated example where the sphere in the boundary of the target plays a role can be constructed via modifying the technique in the examples following Proposition 1.5. What is true, even in one dimension, is that a rational map $f\colon\mathbb{C}^{n}\dashrightarrow\mathbb{C}^{N}$ that restricts to a proper map of $\mathbb{B}_{n}$ to $\mathbb{B}_{N}$ takes all the (nonpole) points of $\mathbb{C}^{n}\setminus\overline{\mathbb{B}_{n}}$ to $\mathbb{C}^{N}\setminus\overline{\mathbb{B}_{N}}$; see Lemma 2.1.

Proper maps of differences of balls are somewhat more complicated. Via a similar argument, again heavily dependent on the result of Forstnerič, we prove the following result.

If $n=1$, the theorem need not hold. For example the function $z\mapsto\frac{1}{z}$ takes an annulus centered at the origin to an annulus, swaps the inside and outside circles and does not extend to the entire disc. The theorem leads us to study what we call rational $m$-fold sphere maps.

In light of this definition, the theorem above says that every proper holomorphic map of ball complements centered at the origin is a rational $2$-fold sphere map. We remark that by Proposition 3.2, we can assume that $R_{1}<R_{2}<\cdots<R_{m}$.

For the tensor of two polynomial maps we find that $\mathopen{}\mathclose{{}\left\lVert f\otimes g}\right\rVert^{2}=\mathopen{}\mathclose{{}\left\lVert f}\right\rVert^{2}\mathopen{}\mathclose{{}\left\lVert g}\right\rVert^{2}$. Hence $\mathopen{}\mathclose{{}\left\lVert z^{\otimes d}}\right\rVert^{2}=\mathopen{}\mathclose{{}\left\lVert z}\right\rVert^{2d}$, and so $z^{\otimes d}$ is a homogeneous $\infty$-fold sphere map. The map $z^{\otimes d}$ does not have linearly independent components for $d\geq 2$, but after applying a unitary and a projection we find a symmetrized homogeneous map $H_{d}$ with linearly independent components such that $\mathopen{}\mathclose{{}\left\lVert z^{\otimes d}}\right\rVert^{2}=\mathopen{}\mathclose{{}\left\lVert H_{d}(z)}\right\rVert^{2}$, where the number of components of $H_{d}$ is the rank of the underlying hermitian form of $\mathopen{}\mathclose{{}\left\lVert z^{\otimes d}}\right\rVert^{2}$. For example in $\mathbb{C}^{2}$, $(z_{1},z_{2})^{\otimes 2}=(z_{1}^{2},z_{1}z_{2},z_{2}z_{1},z_{2}^{2})$ is symmetrized to $H_{2}(z_{1},z_{2})=(z_{1}^{2},\sqrt{2}z_{1}z_{2},z_{2}^{2})$. See D’Angelo [dangelo-1993-several].

By Rudin’s theorem we mentioned above, a homogeneous $1$-fold sphere map is unitarily equivalent to a scalar multiple of $H_{d}$. Thus every homogeneous $1$-fold sphere map is an $\infty$-fold sphere map. Similarly, if we take a direct sum of properly scaled homogeneous sphere maps, we get an $\infty$-fold sphere map. In fact, we have the following:

The question then arises about the existence of other maps than the $\infty$-fold sphere maps. For every $k$ and $m\geq k$, we will show that there exists a rational (nonpolynomial) $k$-fold sphere map of degree $m$ that is not a $(k+1)$-fold sphere map. In particular, every first-degree rational proper map of the difference of zero-centric balls is a unitary composed with an affine linear embedding; however, there exist nonpolynomial second-degree rational maps of a difference of balls. We will also show (Lemma 3.10) that the denominator of a $k$-fold sphere map for $k>1$ must be necessarily of a lower degree than the numerator, extending the analogous result by D’Angelo (see e.g. [dangelo-2019-hermitian]*Proposition 5.1) for sphere maps that also fix the origin.

Finally, we consider proper maps of ball differences to ball complements and vice versa. Using the result of Forstnerič, one can prove that when dimension is at least $2$, no proper holomorphic maps exist between the two different sets.

If $n=1$, start with a proper embedding of the disc into $\mathbb{C}^{N}$ (e.g. Remmert–Bishop–Narasimhan again). We can construct a nonrational proper map from $\mathbb{D}\setminus\overline{r\mathbb{D}}$ to $\mathbb{C}^{N}\setminus\overline{\mathbb{B}_{N}}$: Take a proper holomorphic embedding $f\colon\mathbb{D}\to\mathbb{C}^{N}$. Take a small closed ball $\overline{B}\subset\mathbb{C}^{N}$ so that $f^{-1}(\overline{B})$ is a connected set with more than one point and $\mathbb{D}\setminus f^{-1}(\overline{B})$ is connected. Then it is classical that the doubly connected domain $\mathbb{D}\setminus f^{-1}(\overline{B})$ is biholomorphic to an annulus $\mathbb{D}\setminus\overline{r\mathbb{D}}$ for some $0<r<1$. Composition of the maps and taking $B$ to the unit ball obtains the desired map. To construct the second map when $n=1$, $\mathbb{C}\setminus\overline{\mathbb{D}}$ is biholomorphic to the punctured disc $\mathbb{D}^{*}$, which can be properly embedded into some $\mathbb{B}_{N}\setminus\overline{B_{R}(C)}$ in many ways (e.g., linearly).

Interestingly, it is not difficult to construct many nontrivial proper maps of $\mathbb{B}_{n}$ to $\mathbb{C}^{N}\setminus\overline{\mathbb{B}_{N}}$ when $N>n$, but the proposition says that if $n\geq 2$, there is no way to properly map the annulus to the complement of the ball.

The organization of this paper is the following. In section 2, we study proper maps of ball complements and prove Theorem 1.1. In section 3, we study the rational $m$-fold sphere maps and prove Theorem 1.4. Finally, in section 4, we study proper mappings of ball differences and we prove Theorem 1.2.

We start with a lemma about where rational proper holomorphic maps of balls take the complement of the ball. We remark that unlike many of the results we consider, this lemma still holds in $n=1$.

The lemma proves the converse statement of Theorem 1.1. That is, if $f\colon\mathbb{C}^{n}\to\mathbb{C}^{N}$ is a polynomial map such that its restriction to $\mathbb{B}_{n}$ is a proper map to $\mathbb{B}_{N}$, it takes the sphere $S^{2n-1}$ to the sphere $S^{2N-1}$, and the lemma says that it takes $\mathbb{C}^{n}\setminus\overline{\mathbb{B}_{n}}$ to $\mathbb{C}^{N}\setminus\overline{\mathbb{B}_{N}}$. If we furthermore assume that $\mathopen{}\mathclose{{}\left\lVert f(z)}\right\rVert\to\infty$ as $\mathopen{}\mathclose{{}\left\lVert z}\right\rVert\to\infty$, we find that $f$ is proper.

It is not clear if the norm of a polynomial proper of balls always goes to infinity at infinity thereby giving a proper map of $\mathbb{C}^{n}\setminus\overline{\mathbb{B}_{n}}$ to $\mathbb{C}^{N}\setminus\overline{\mathbb{B}_{N}}$. We provide a proof in some natural special cases.

We remark that the conclusion of the proposition also follows if the polynomial proper map of balls is constructed using tensoring only starting with the identity in the procedure of D’Angelo [dangelo-1988-proper].

The rest of Theorem 1.1 follows from the following lemma.

This section is split into two parts. In the first part, we consider polynomial $m$-fold sphere maps, proving Theorem 1.4 for polynomials. We then give a construction of maps of higher degree showing that the bound is sharp. In the second part of the section, we extend the results to rational maps and give the construction showing that the result is sharp also in the rational case.