Finiteness theorems for holomorphic mappings from products of hyperbolic Riemann surfaces

TL;DR

This paper proves that the set of non-constant holomorphic maps from products of hyperbolic Riemann surfaces to certain hyperbolic manifolds is finite, highlighting a finiteness property in complex geometry.

Contribution

It establishes finiteness theorems for holomorphic mappings from products of hyperbolic Riemann surfaces into hyperbolic manifolds with bounded universal cover.

Findings

The space of such holomorphic mappings is finite.

Finiteness holds for dominant and non-constant mappings.

Results apply to hyperbolic Riemann surfaces of finite type.

Abstract

We prove that the space of dominant/non-constant holomorphic mappings from a product of hyperbolic Riemann surfaces of finite type into certain hyperbolic manifolds with universal cover a bounded domain is a finite set.

Equations

$$\sum _{g\in G}(1-|g(z)|)=+\infty \forall z\in \mathbb{D}.$$

$$U_1\supset U_2\supset \dots ,$$

$$U_1\supset U_2\dots ,$$

$$\underset{z\to \xi }{\lim }f(z),$$

$$\tilde{\varphi }\circ g$$

$$\tilde{\psi }\circ g$$

$$\underset{k\to \infty }{\lim }\tilde{\varphi }\circ g_k(z)$$

$$\underset{k\to \infty }{\lim }\tilde{\psi }\circ g_k(z)$$

Full text

1. Introduction

Complex analysis, in its most general setting, is the study of holomorphic mappings between complex spaces. Therefore, it is somewhat paradoxical that the space $\mathscr{O}(X,Y)$ of holomorphic mappings between complex spaces $X$ and $Y$ often comprises just constant mappings or contains only a finite/discrete set of non-constant mappings. The classical theorems of Liouville and Picard illustrate this phenomenon. In the context of compact Riemann surfaces, the famous Riemann–Hurwitz formula puts severe restrictions on the comparative genera of the Riemann surfaces $R$ and $S$ whenever $\mathscr{O}_{*}(R,S)$ is non-empty. A somewhat deeper result is the following result of Imayoshi, which is an extension of a classical theorem due to de Franchis [dF13].

Result 1** (Imayoshi [Ima83, Theorem 2]).**

Let $R$ be a Riemann surface of finite type and let $S$ be a Riemann surface of finite type $(g,n)$ with $2g-2+n>0$. Then $\mathscr{O}_{*}(R,S)$ is a finite set.

A Riemann surface $S$ is said to be of finite type $(g,n)$ if it is biholomorphic to a Riemann surface obtained by removing $n$ points from a compact Riemann surface of genus $g$. Observe that the condition $2g-2+n>0$ ensures that $S$ is a hyperbolic Riemann surface. This fact is crucial in the proof. Also note that a hyperbolic Riemann surface is automatically complete Kobayashi hyperbolic. However, aside from an appeal to Montel’s theorem for bounded domains, Kobayashi hyperbolicity plays no role in Imayoshi’s proof. Rather, the proof relies mainly on the theory of Fuchsian groups. It is well-known that the presence of (Kobayashi) hyperbolicity in either the domain or target complex space often forces the space of dominant holomorphic mappings to be finite/discrete. See [KO75] and [NS82] for examples of such results. Also see [ZL89, Chapter 2] for a detailed survey.

The following result of Shiga can be thought of as a higher dimensional analogue of Result 1.

Result 2** (Shiga [Shi04, Theorem 1.3]).**

Let $X=\mathbb{B}^{n}/G$ be a complex hyperbolic manifold of divergence type and let $Y=\Omega/\Gamma$ be a geometrically finite $n$-dimensional complex manifold where $\Omega\subset\mathbb{C}^{m}$ is a bounded domain and $\Gamma$ is fixed-point-free discrete subgroup of $\textsf{Aut}(\Omega)$. Suppose $G$ is finitely generated and that $\Omega$ is complete with respect to the Kobayashi distance. Then $\mathscr{O}_{\textsf{dom}}(X,Y)$ is a finite set. Furthermore, if the essential boundary dimension of $\Omega$ is zero, then $\mathscr{O}_{*}(X,Y)$ is a finite set.

Remark 3*.*

The proof of the above result uses methods inspired by the ones used by Imayoshi. However, as is evident from the hypotheses, Kobayashi hyperbolicity plays a prominent role in the proof.

The purpose of this work is to both extend Result 1 and also to clarify the role played by Kobayashi hyperbolicity. The central result of this paper is:

Theorem 4**.**

Let $X:=X_{1}\times\dots\times X_{n}$ be a product of hyperbolic Riemann surfaces of finite type and let $Y=\Omega/\Gamma$ be an $m$-dimensional complex manifold where $\Omega\subset\mathbb{C}^{m}$ is a bounded domain and $\Gamma$ is fixed-point-free discrete subgroup of $\textsf{Aut}(\Omega)$.

1. (1)

If $N$ is a tautly embedded complex submanifold of $Y$ then $\mathscr{O}_{\textsf{dom}}(X,N)$ is a finite set. 2. (2)

If $Y$ is geometrically finite and $\Omega$ is complete hyperbolic then $\mathscr{O}_{\textsf{dom}}(X,Y)$ is a finite set. 3. (3)

If in addition to the conditions in (2), the essential boundary dimension of $\Omega$ is zero, then $\mathscr{O}_{*}(X,Y)$ is a finite set.

Remark 5*.*

This theorem subsumes Result 1 (by Remark 16 and (2) above) and can be viewed as an analogue of Result 2 with $\mathbb{B}^{n}$ replaced by $\mathbb{D}^{n}$. Our proof uses a combination of techniques used in the proofs of Result 1 and 2. However, a key point of departure is our use of a normal families argument (see Lemma 24) which simplifies our proof. Our proof is also intentionally written in such a way that we get new simplified proofs of Result 1 and Result 2 mutatis mutandis.

Notation: We shall denote the space of holomorphic mappings between two complex spaces $X$ and $Y$ by $\mathscr{O}(X,Y)$. The space of non-constant holomorphic mappings will be denoted by $\mathscr{O}_{*}(X,Y)$ and dominant holomorphic mappings by $\mathscr{O}_{\textsf{dom}}(X,Y)$. $\mathbb{D}$ shall denote the open unit disk in the complex plane.

Organization: In Section 2, we summarize (with references) the results and tools needed in the proof of Theorem 4. The proof of Theorem 4 forms the content of Section 3. Finally we present a few applications of our results in Section 4.

2. Summary of results and tools needed

2.1. Kobayashi hyperbolicity

On each complex manifold $M$, one can define a pseudodistance $K_{M}$, called the Kobayashi pseudodistance, that is distance decreasing under holomorphic mappings, i.e., if $f:M\to N$ is a holomorphic mapping between complex manifolds, then $K_{M}(x,y)\geq K_{N}(f(x),f(y))\ \forall x,y\in M$. By definition, the Kobayashi pseudodistance on $\mathbb{D}$ is nothing but the distance obtained by integrating the Poincaré metric. See [Kob67] for basic facts about the Kobayashi pseudodistance and [Kob98, Kob05] for a detailed exposition.

We say that a complex manifold $M$ is hyperbolic if $K_{M}$ is a distance. If in addition $(M,K_{M})$ is a complete metric space, we say that $M$ is complete Kobayashi hyperbolic. These notions have been extensively studied in the literature and are indispensable in the study of holomorphic mapping.

The following result is often used to determine if a given complex manifold is (complete) hyperbolic.

Result 6**.**

Let $M$ and $N$ be complex manifolds and $F:M\to N$ a covering map. Then $M$ is (complete) hyperbolic iff $N$ is.

If $M$ and $N$ are hyperbolic complex manifolds then $\mathscr{O}(M,N)$ is obviously an equicontinuous family under the Kobayashi distance. It is a natural question whether $\mathscr{O}(M,N)$ is relatively compact as a subspace of $\mathcal{C}(M,N)$ in the compact-open topology. As the compact-open topology coincides with topology of uniform convergence on compacts, the following definition is natural.

Definition 7**.**

Let $M$ and $N$ be complex manifolds. A subset $\mathcal{F}\subset\mathcal{C}(M,N)$ is said to be a normal family if every sequence $\{f_{n}\}\subset\mathcal{F}$ has either a subsequence that converges uniformly on compacts to a function in $\mathcal{C}(M,N)$ or has a compactly divergent subsequence.

A complex manifold $N$ is said to be taut if for every complex manifold $M$ the set $\mathscr{O}(M,N)$ is a normal family.

The following result relates the notion of hyperbolicity and tautness.

Result 8** (Kiernan [Kie70, Proposition 3]).**

Complete hyperbolic complex manifolds are taut.

2.2. Groups of divergence type

The classical uniformization theorem states that the universal cover of a Riemann surface is either $\mathbb{C},\widehat{\mathbb{C}}$ or $\mathbb{D}$. We say that a Riemann surface is hyperbolic if its universal cover is $\mathbb{D}$. By Result 6 a hyperbolic Riemann surface is automatically complete Kobayashi hyperbolic.

If $R$ is a hyperbolic Riemann surface, we may write $R$ as $\mathbb{D}/G$, where $G$ is a fixed-point free discrete subgroup of $\textsf{Aut}(\mathbb{D})$. Such a group $G$ is knows as a Fuchsian group and the representation of $R$ as $\mathbb{D}/G$ is known as the Fuchisan group representation of $R$. From the theory of covering spaces, it also follows that $G\simeq\pi_{1}(R)$. If $R$ is a hyperbolic Riemann surface of finite type, we can get considerable information on the action of $G$ on $\mathbb{D}$. We first need a definition.

Definition 9**.**

A Fuchsian group $G$ is said to be of divergence type if

[TABLE]

Result 10**.**

Let $R:=\mathbb{D}/G$ be a hyperbolic Riemann surface. Then the following are equivalent

1. (1)

$G$* is a Fuchsian group of divergence type.* 2. (2)

$R$* admits no Green’s function.* 3. (3)

For almost every $\xi\in\partial\mathbb{D}$, we can find a sequence $g_{\nu}\in G$ such that $g_{\nu}\to\xi\ \forall z\in\mathbb{D}$ uniformly and non-tangentially on compacts.

Remark 11*.*

It follows from the above result that hyperbolic Riemann surfaces of finite type can be represented as $\mathbb{D}/G$ where $G$ is a Fuchsuan group of divergence type. See [Tsu75] for a proof of the above result and related results.

2.3. The notion of ends and essential boundary dimension

Definition 12**.**

Let $X$ be a topological manifold. An end of $X$ is a decreasing sequence of connected open sets

[TABLE]

with the property that given any compact set $K\subset X$, there is an $N_{0}\in\mathbb{Z}_{+}$ such that $K\cap U_{N_{0}}=\emptyset$. Two ends $U_{1}\subset U_{2}\subset\dots$ and $U_{1}^{\prime}\subset U_{2}^{\prime}\subset\dots$ are considered equivalent if for each $n\in\mathbb{Z}_{+}$, we can find $N\in\mathbb{Z}_{+}$ such that $U_{n}\subset U_{N}^{\prime}$, and vice-versa.

Remark 13*.*

The notion of ends can be used to construct a nice compactification of $X$. The space of ends also allows one to classify non-compact surfaces. See [Ric63].

In this work, we are mainly interested in complex manifolds with finitely many ends of a certain kind known as parabolic ends. We first define the notion of a pluripolar set.

Definition 14**.**

A set $E\subset\mathbb{C}^{n}$ is said to be a complete pluripolar set if for some plurisubharmonic function $u:\mathbb{C}^{n}\to[-\infty,\infty),E=u^{-1}\{-\infty\}$.

We now define what it means for a manifold $Y:=\Omega/\Gamma$ to be geometrically finite.

Definition 15**.**

Let $Y:=\Omega/\Gamma$ be a complex manifold, where $\Omega\subset\mathbb{C}^{n}$ is a bounded domain and $\Gamma$ is a discrete and fixed-point free subgroup of $\textsf{Aut}(\Omega)$. Denote the covering map by $\pi:\Omega\to X$. We say that the end

[TABLE]

is a parabolic end if for some $N_{0}$, we can find a connected component of $\pi^{-1}(U_{0})$, say $\widetilde{U}_{N_{0}}$, such that $\overline{\widetilde{U}}_{N_{0}}\cap\partial\Omega\subset\bigcup_{j=1}^{\infty}R_{j}$ where each $R_{j}$ is a complete pluripolar set in $\mathbb{C}^{n}\setminus\Omega$.

We say that $X$ is geometrically finite if $X$ has only finitely many ends and each end is a parabolic end.

Remark 16*.*

Any hyperbolic Riemann surface of finite type is automatically geometrically finite.

Remark 17*.*

If a manifold $X$ has only finitely many inequivalent ends then we may, and shall do so tacitly, represent each of these finitely many ends by finitely many pairwise disjoint connected open sets.

2.4. Fatou–Riesz theorems on the polydisk

We will need the following version of the Fatou–Riesz theorem for the polydisk.

Result 18** (Tsuji [Tsu45]).**

Let $f$ be a bounded holomorphic function on $\mathbb{D}^{n}$. Then for almost every point $\xi$ on the distinguished boundary (the torus $\mathbb{T}^{n}$)

[TABLE]

exists whenever $z\to\xi$ non-tangentially. We denote this limit by $f^{*}(\xi)$. Furthermore, if $g$ is another bounded holomorphic function on $\mathbb{D}^{n}$ and $f^{*}(\xi)=g^{*}(\xi)$ on a set of positive measure on $\mathbb{T}^{n}$, then $f\equiv g$.

The following lemma has been proven for an arbitrary bounded $\mathcal{C}^{2}$-smooth domains by Shiga (see [Shi04, Theorem 3.1]). We can view it as a generalization of the last part of the previous theorem. As the proof given by Shiga relies only on the existence of a Poisson kernel, his proof works for the polydisk also.

Lemma 19**.**

Let $\phi:\mathbb{D}^{n}\to\mathbb{C}^{m}$ be a non-constant bounded holomorphic mapping and let $E$ be a complete pluripolar set in $\mathbb{C}^{m}$. If $\phi^{*}(\xi)\in E$ for a set of positive measure on $\mathbb{T}^{n}$, then $\phi(\mathbb{D}^{n})\subset E$.

We will also need the following lemma which shows that under the hypothesis of Theorem 4, the non-tangential limits of the lift of a non-constant map lie on the boundary. The proof is essentially the same as the proof of [Shi04, Lemma 2.2].

Lemma 20**.**

Let $X=X_{1}\times\dots\times X_{n}$ be a product of hyperbolic Riemann surfaces of finite type and let $Y:=\Omega/\Gamma$ be a complex manifold, where $\Omega\subset\mathbb{C}^{n}$ is a bounded domain and $\Gamma$ is a discrete and fixed-point free subgroup of $\textsf{Aut}(\Omega)$. Let $f:X\to Y$ be a non-constant holomorphic map and let $F:\mathbb{D}^{n}\to\Omega$ be any lift. Then for almost every $\xi\in\mathbb{T}^{n},F^{*}(\xi)\in\partial\Omega$.

We require one more definition.

Definition 21**.**

Let $D\subset\mathbb{C}^{n}$ be a bounded domain. We say that $D$ has essential boundary dimension zero, if there exists a family $\{R_{j}\}_{j=1}^{\infty}$ of complete pluripolar sets with $R_{j}\cap\partial D\neq\emptyset\ \forall j$ such that $\partial D\setminus\bigcup_{j=1}^{\infty}R_{j}$ contains no analtic space of positive dimension.

3. Proofs of our main results

In this section, we shall give the proof of our main theorem (Theorem 4). We first prove a rigidity theorem for holomorphic mappings (Theorem 22). This rigidity theorem is an extension of a rigdity result used by Imayoshi to prove Result 1.

Theorem 22**.**

Let $X:=X_{1}\times\dots\times X_{n}$ be a product of hyperbolic Riemann surfaces of finite type and let $Y=\Omega/\Gamma$ be an $m$-dimensional complex manifold where $\Omega\subset\mathbb{C}^{m}$ is a bounded domain and $\Gamma$ is fixed-point-free discrete subgroup of $\textsf{Aut}(\Omega)$. Let $\phi,\psi:X\to Y$ be holomorphic mappings such that $\phi_{*}=\psi_{*}$. Then

1. (1)

If $\phi$ (or $\psi$) is dominant, then $\phi=\psi$. 2. (2)

If $\phi$ (or $\psi$) is non-constant and $\Omega$ has essential boundary dimension zero, then $\phi=\psi$.

Remark 23*.*

The above theorem is also true when $X=\mathbb{B}^{n}/G$ (see Section 4). Our proof works mutatis mutandis. See also [Shi04, Theorem 1.1].

Proof.

Let $G_{i}$ be the Fuchsian group of divergence type such that $X_{i}=\mathbb{D}/G_{i}$. Then $X=\mathbb{D}^{n}/G$ where $G:=\bigoplus_{i=1}^{n}G_{i}$. Let $\widetilde{\phi},\widetilde{\psi}:\mathbb{D}^{n}\to\Omega$ be lifts of $\phi$ and $\psi$, respectively. As $\mathbb{D}^{n}$ is simply-connected, the group $G$ and $\pi_{1}(X)$ are isomorphic. Fix a point $y\in Y$ and a point $\tilde{y}\in\Omega$ that gets mapped to $y$ under the quotient map. Then, given an element $[\delta]\in\pi_{1}(Y,y)$ there is an element $\gamma\in\Gamma$ that takes $\tilde{y}$ to the end point of the lift of $\delta$ starting at $\tilde{y}$. This element is independent of the choice of representative of $[\delta]$. It is easy to check that, in this way, we get a homomorphism from $\pi_{1}(Y)$ to $\Gamma$. Thus, given any map from $X$ to $Y$, by composing, we get a homomorphism from $G$ to $\Gamma$. The hypothesis that $\phi_{*}=\psi_{*}$ implies that both mappings $\phi$ and $\psi$ induce the same homomorphism from $G$ to $\Gamma$. Let $\chi$ be this induced homomorphism.

It follows that

[TABLE]

for each $g\in G$.

As each $G_{i}$ is of divergence type, by Result 10, for almost every $\xi\in\mathbb{T}^{n}$, we can find a sequence $g_{i,k}\in G_{i}$ such that $g_{i,k}(z)\to\xi_{i}$ non-tangentially on compact subsets of $\mathbb{D}$. Let $g_{k}:=(g_{1,k},g_{2,k},\dots,g_{n,k})$. Then $g_{k}(z)\to\xi$ non-tangentially and uniformly on compact subsets of $\mathbb{D}^{n}$.

Now, as $\Omega$ is a bounded domain, Fatou’s theorem for the polydisk (Result 18) implies that for almost every $\xi\in\mathbb{T}^{n}$, each component function of $\widetilde{\phi}$ and $\widetilde{\psi}$ converge as $z\to\xi$ non-tangentially. We define $\tau(\xi)$ and $\eta(\xi)$ to be the non-tangential limits of $\widetilde{\phi}$ and $\widetilde{\psi}$, respectively, as $z\to\xi$ (the functions $\tau$ and $\psi$ are defined almost-everywhere on $\mathbb{T}^{n}$). This, in combination with arguments in the previous paragraph, shows that for almost every $\xi\in\mathbb{T}^{n}$, we can find a sequence $g_{k}\in G$ such that

[TABLE]

for each $z\in\mathbb{D}^{n}$. As each $\chi(g_{k})\in\textsf{Aut}({\Omega})$, and as $\Omega$ is bounded, we may use Montel’s theorem and assume without loss of generality that $\chi(g_{k})$ converges uniformly on compact subsets of $\Omega$ to a holomorphic map $A_{\xi}:\Omega\to\mathbb{C}^{n}$. Using (3.1) and (3.2), we have $A_{\xi}\circ\widetilde{\phi}(z)=\tau(\xi)$ and $A_{\xi}\circ\widetilde{\psi}(z)=\eta(\xi)$, whenever $z\in\mathbb{D}^{n}$.

Now assume that $\phi$ is dominant. This implies that $\widetilde{\phi}$ is dominant as well. Therefore, we can find a polydisk $P_{\xi}\Subset\mathbb{D}^{n}$ such that $\widetilde{\phi}(P_{\xi})$ has an interior point. This forces $A_{\xi}$ to be a constant map by the identity principle. Hence $\tau$ and $\eta$ agree almost everywhere on the torus which means that $\widetilde{\phi}=\widetilde{\psi}$ (by Fatou’s theorem) and hence $\phi=\psi$ as claimed.

Now assume that the essential boundary dimension of $\Omega$ is [math] and that $\phi$ is non-constant. This implies that $\widetilde{\phi}$ is non-constant as well. Each $\chi(g_{k})$ is an automorphism of the bounded domain $\Omega$ that converges in the compact-open topology to the map $A_{\xi}:\Omega\to\overline{\Omega}$. Consequently, by Cartan’s theorem for biholomorphisms, either $A_{\xi}$ is an automorphism or the image of $A_{\xi}$ is fully contained in $\partial\Omega$. However, as $\widetilde{\phi}$ is non-constant, $\widetilde{\phi}(P_{\xi})$ cannot be a singleton set (by the identity principle), where $P_{\xi}\Subset\mathbb{D}^{n}$ is any polydisk. Hence $A_{\xi}$ is not injective as $A_{\xi}\circ\widetilde{\phi}(z)$ is constant and consequently $A_{\xi}(\Omega)\subset\partial\Omega$.

We have shown that for almost every $\xi\in\mathbb{T}^{n}$, we have $A_{\xi}(\Omega)\subset\partial\Omega$. We claim that for almost every $\xi\in\mathbb{T}^{n}$, $A_{\xi}$ is constant. Suppose not. Since the essential boundary dimension of $\Omega$ is [math], we can find countably many complete pluripolar sets $R_{j}$ of $\mathbb{C}^{n}\setminus\Omega$ such that the image of any non-constant holomorphic function into $\partial\Omega$ must lie fully in $\bigcup_{j=1}^{\infty}R_{j}$. Therefore, we can find a set $E\subset\mathbb{T}^{n}$ of positive measure with the property that $A_{\xi}(\Omega)\subset R_{j_{0}}$ for some $j_{0}\in\mathbb{Z}_{+}$ and $\forall\xi\in E$. This means that for a set of of positive measure, the non-tangential limits of $\widetilde{\phi}$ lie in a complete pluripolar set and consequently $\widetilde{\phi}(\mathbb{D}^{n})$ must be entirely contained in the same pluripolar set by Lemma 19. This is not possible. Consequently, $A_{\xi}$ is a constant for almost every $\xi\in\mathbb{T}^{n}$. This means that the non-tangential limits of $\widetilde{\phi}$ and $\widetilde{\psi}$ coincide almost everywhere on $\mathbb{T}^{n}$ proving that $\widetilde{\phi}=\widetilde{\psi}$ by Fatou’s theorem. Hence $\phi=\psi$.

∎

The next lemma clarifies the role of geometrical finiteness of $\Omega$.

Lemma 24**.**

Let $X:=X_{1}\times\dots\times X_{n}$ be a product of hyperbolic Riemann surfaces of finite type and let $Y=\Omega/\Gamma$ be an $m$-dimensional geometrically finite complex manifold where $\Omega\subset\mathbb{C}^{m}$ is a bounded complete hyperbolic domain and $\Gamma$ is fixed-point-free discrete subgroup of $\textsf{Aut}(\Omega)$. Then any sequence $\{f_{n}\in\mathscr{O}_{*}(X,Y)\}$ converges in the compact-open topology to a map in $\mathscr{O}(X,Y)$.

Proof.

As in the proof of Theorem 22, we write $X$ as $\mathbb{D}^{n}/G$.

By Result 8, complete hyperbolic manifolds are taut. Therefore any sequence $\{f_{\nu}\in\mathscr{O}_{*}(X,Y)\}$ must be either have a compactly convergent or compactly divergent subsequence.

Suppose there is a sequence $\{f_{\nu}\in\mathscr{O}_{*}(X,Y)\}$ that is compactly divergent. We view each $X_{i}$ as being an open subset of a compact Riemann surface $R_{i}$. For each point $x\in R_{i}\setminus X_{i}$, let $D_{x}$ be a small disk in $R_{i}$ with $x$ as the centre, chosen in such a way that if $x_{1},x_{2}\in R_{i}\setminus X_{i}$ are distinct points, then the closures of $D_{x_{1}}$ and $D_{x_{2}}$ are disjoint. Let $K:=\bigtimes_{i=1}^{n}R_{i}\setminus\bigcup_{x\in R_{i}\setminus X_{i}}D_{x}$. As $K$ is a deformation retract of $X$, $\pi_{1}(K)=\pi_{1}(X)$. Any curve $\gamma\subset X$ can be homotoped to a curve that lies entirely in $K$. For large $\nu$, by compact divergence and the fact that $K$ is connected, we may assume that $f_{\nu}(K)\subset V$ where $V\subset Y$ is one of the finitely many parabolic ends of $Y$. Let $V^{\prime}$ be a connected component of a pre-image of $V$ under the covering map such that $\overline{V^{\prime}}\cap\partial\Omega\subset\bigcup_{i=1}^{\infty}R_{i}$ where each $R_{j}$ is a complete pluripolar set in $\mathbb{C}^{m}\setminus\Omega$.

Fix $\nu$ large and let $f:=f_{\nu}$ and let $\xi:G\to\Gamma$ be the induced homomorphism. Denote by $F$ the lift of $f$ with the property that $F(0)\in V^{\prime}$. Let $p_{0}\in M$ be the image of [math] under the covering map. We may assume that $p_{0}\in K$. Let $g\in G$ and let $L$ be a loop based at $p_{0}$ contained entirely in $K$ that corresponds to $g$. Denote the lift of $L$ based at [math] by $\widetilde{L}$. Then $F(\widetilde{L})\subset V^{\prime}$. As $F(g(0))=\chi(g)(F(0))$ and the endpoint of $\widetilde{L}$ is $g(0)$, it follows that $\chi(g)(F(0))\in V^{\prime}$. For almost every $\xi\in\mathbb{T}^{n}$, we can find a sequence $g_{\nu}\in G$ such that $g_{\nu}(0)\to\xi$ non-tangentially. We assume that $F^{*}(\xi)$ exists and that $F^{*}(\xi)\in\partial\Omega$. But $F(g_{\nu}(0))=\chi(g_{\nu})(F(0))\in V^{\prime}$. Hence $F^{*}(\xi)\in\overline{V^{\prime}}\cap\partial\Omega\subset\bigcup_{i=1}^{\infty}R_{i}$. This force $F$ to be a constant by Lemma 19. ∎

Proof of Theorem 4: Let $\{f_{k}\}\subseteq\mathscr{O}_{\textsf{dom}}(X,N)$ be a sequence of distinct dominant holomorphic mappings. As $N$ is tautly embedded in $Y$, we may assume that the sequence $\{f_{k}\}$ converges in the compact-open topology to a map $f:X\to Y$.

As in the proof of Lemma 24, we can find a connected and compact set $K$ such that finitely many closed loops contained in $K$ generate $\pi_{1}(X)$. Now, as $\{f_{k}\}$ converges uniformly on $K$ to $f$, it is easy to see that for suitably large $k$, the $f_{k}(\gamma)$ and $f(\gamma)$ are homotopic where $\gamma$ is curve whose image lies entirely in $K$. Thus all the $f_{k}$’s induce the same homomorphism on $\pi_{1}(X)$. Consequently, by Theorem 22, for large $k$, all the $f_{k}$’s are equal, which is a contradiction.

We prove the last part. Suppose $\{f_{k}\}\subseteq\mathscr{O}_{*}(X,Y)$ is a sequence of distinct holomorphic mappings. By Lemma 24, we may, by passing to a subsequence, if needed, assume that the $f_{k}$’s converge uniformly on compacts. The previous argument now delivers the proof.

Part 2 of this theorem follows by similar arguments. ∎

4. Applications

We will now highlight a few applications of our results. The first application is rather trivial but nevertheless interesting.

Corollary 25**.**

Let $X:=X_{1}\times\dots\times X_{n}$ be a product of hyperbolic Riemann surfaces of finite type and let $Y=\Omega/\Gamma$ be an $m$-dimensional compact complex manifold where $\Omega\subset\mathbb{C}^{m}$ is a bounded domain and $\Gamma$ is fixed-point-free discrete subgroup of $\textsf{Aut}(\Omega)$ Then $\mathscr{O}_{\textsf{dom}}(X,N)$ is a finite set where $N\subset Y$ is any complex submanifold (including $Y$ itself).

Proof.

As $Y$ is compact hyperbolic, it is complete hyperbolic and therefore taut. Any sequence of holomorphic maps $f_{k}:X\to Y$ cannot have a compactly divergent subseqeunce as $Y$ is compact. Thus $N$ is tautly embedded in $Y$ and the corollary follows from Theorem 4. ∎

Our next application is an extension of Theorem 1 to maps between products of Riemann surfaces.

Corollary 26**.**

Let $X=X_{1}\times\dots\times X_{n}$ be a product of hyperbolic Riemann surfaces of finite type and let $Y=Y_{1}\times\dots\times Y_{m}$ be a product of hyperbolic Riemann surfaces each of which can be embedded inside a compact Riemman surface. Then $\mathscr{O}_{\textsf{dom}}(X,Y)$ is a finite set.

Remark 27*.*

In [Jan14], finite proper holomorphic mappings between products of certain hyperbolic Riemann surfaces was studied. The above result is a partial extension of the main result there.

Proof.

We view each $Y_{i}$ as an open subset of a compact Riemann surface $S_{i}$. Now, $Y_{i}$ is obtained by excising a set $E_{i}$ from $S_{i}$. Choose finite sets $E_{i}^{\prime}\subset E_{i}$ such that $S_{i}\setminus E_{i}^{\prime}$ is hyperbolic. Let $N:=S_{1}\setminus E_{1}^{\prime}\times\dots\times S_{n}\setminus E_{n}^{\prime}$. Then $N$ is a product of hyperbolic Riemann surfaces of finite type. It suffices to show that $\mathscr{O}_{\textsf{dom}}(X,N)$ is a finite set. But any hyperbolic Riemann surface of finite type is geometrically finite. It follows from Theorem 4 that $\mathscr{O}_{\textsf{dom}}(X,N)$ is a finite set. ∎

We end with an observation about our results. We used three features of the representation of the manifold $X$ as $\mathbb{D}^{n}/G$ in our results:

1. (1)

We have a Fatou–Riesz theorem on $\mathbb{D}^{n}$. 2. (2)

For almost every element $\xi\in\mathbb{T}^{n}$, we have a sequence of elements $g_{k}\in G$ such that $g_{k}$ converges uniformly and non-tangentially on compacts to $\xi$. In other words, $G$ is of divergence type. 3. (3)

$G$ (or equivalently $\pi_{1}(X)$) is finitely generated.

In view of the above observation, our proof of Theorem 4 works mutatis mutandis when $X$ is of the form $\mathbb{B}^{n}/G$ where $G$ is a finitely generated subgroup of $\textsf{Aut}(\mathbb{B}^{n})$ of divergence type. See [Shi04] for a precise definition of divergence type. We can also replace $X$ by $D/G$ where $D$ is any bounded domain and group $G\subset\textsf{Aut}(D)$ that have the analogues of the three features mentioned above. For instance, in the statement of Theorem 4, we may replace each Riemann surface $X_{i}$ by a quotient $\mathbb{B}^{n}/G_{i}$ where each $G_{i}\subset\textsf{Aut}(\mathbb{B}^{n})$ is a finitely generated subgroup of divergence type.