A Metric Version of Poincaré’s Theorem Concerning Biholomorphic Inequivalence of Domains

Abstract

We show that if $Y_j\subset \mathbb {C}^{n_j}$ is a bounded strongly convex domain with $C^3$-boundary for $j=1,\dots ,q$, and $X_j\subset \mathbb {C}^{m_j}$ is a bounded convex domain for $j=1,\ldots ,p$, then the product domain $\prod _{j=1}^p X_j\subset \mathbb {C}^m$ cannot be isometrically embedded into $\prod _{j=1}^q Y_j\subset \mathbb {C}^n$ under the Kobayashi distance, if $p>q$. This result generalises Poincaré’s theorem which says that there is no biholomorphic map from the polydisc onto the Euclidean ball in $\mathbb {C}^n$ for $n\ge 2$. The method of proof only relies on the metric geometry of the spaces and will be derived from a more general result for products of proper geodesic metric spaces with the sup-metric. In fact, the main goal of the paper is to establish a general criterion, in terms of certain asymptotic geometric properties of the individual metric spaces, that yields an obstruction for the existence of an isometric embedding between product metric spaces.

1 Introduction

Numerous theorems in several complex variables are instances of results in metric geometry. In this paper, we shall see that a classic theorem due to Poincaré [22], which says that there is no biholomorphic map from the polydisc $\Delta ^n$ onto the (open) Euclidean ball $B_n$ in $\mathbb {C}^n$ if $n\ge 2$, is a case in point. In fact, it is known [19, 29, 30] that there exists no surjective Kobayashi distance isometry of $\Delta ^n$ onto $B_n$ if $n\ge 2$. More generally, one may wonder when it is possible to isometrically embed a product domain $\prod _{j=1}^p X_j\subset \mathbb {C}^m$ into another product domain $\prod _{j=1}^q Y_j\subset \mathbb {C}^n$ under the Kobayashi distance. In this paper, we show, among other results, the following theorem.

Theorem 1.1

Suppose that $X_j\subset \mathbb {C}^{m_j}$ is a bounded convex domain for $j=1,\ldots ,p$, and $Y_j\subset \mathbb {C}^{n_j}$ is a bounded strongly convex domain with $C^3$-boundary for $j=1,\dots ,q$.. If $p>q$, then there is no isometric embedding of $\prod _{j=1}^p X_j$ into $\prod _{j=1}^q Y_j$ under the Kobayashi distance.

Note that Poincaré’s theorem is a special case where $p=n\ge 2$ and $q=1$, as the boundary of the Euclidean ball is smooth. The case where $\sum _j m_j = \sum _j n_j$ and the isometry is surjective was analysed by Zwonek [29,  Theorem 2.2.5] who used different methods.

A key property of the Kobayashi distance is the product property, see [13, Theorem 3.1.9]. Indeed, if $X_j\subset \mathbb {C}^{m_j}$ is a bounded convex domain for $j=1,\ldots ,p$, then the Kobayashi distance, $k_X$, on the product domain $X:=\prod _{j=1}^p X_j$ satisfies

$$\begin{aligned} k_X(w,z) =\max _{j=1,\ldots ,p} k_{X_j}(w_j,z_j) \quad \hbox {for all }w=(w_1,\ldots ,w_p),z=(z_1,\ldots ,z_p)\in X. \end{aligned}$$

In view of the product property, it is natural to consider product metric spaces with the sup-metric. Given metric spaces $(M_j,d_j)$, $j=1,\ldots ,p$, the product metric space $(\prod _{j=1}^p M_j,d_\infty )$ is given by

$$\begin{aligned} d_\infty (x,y) := \max _j d_j(x_j,y_j) ~\hbox {for} ~x=(x_1,\ldots ,x_p),y=(y_1,\ldots ,y_p)\in \prod _{j=1}^p M_j. \end{aligned}$$

In this general context, it is interesting to understand when one can isometrically embed a product metric space into another one. The main goal of this paper is to establish a general criterion, in terms of certain asymptotic geometric properties of the individual metric spaces, that yields an obstruction for the existence of an isometric embedding between product metric spaces, and to show how this criterion can be used to derive Theorem 1.1.

The key concepts from metric geometry involved are as follows: the horofunction boundary of proper geodesic metric spaces, almost geodesics, Busemann points, the detour distance, and the parts of the horofunction boundary, which will all be recalled in the next section. Our main result is the following.

Theorem 1.2

Suppose that $(M_j,d_j)$ is a proper geodesic metric space containing an almost geodesic sequence for $j=1,\ldots ,p$, and $(N_j,\rho _j)$ is a proper geodesic metric space such that all its horofunctions are Busemann points, and $\delta (h_j,h_j')=\infty$ for all $h_j\ne h_j'$ Busemann points of $(N_j,\rho _j)$, for $j=1,\ldots ,q$. If $p>q$, then there exists no isometric embedding of $(\prod _{j=1}^p M_j,d_\infty )$ into $(\prod _{j=1}^qN_j, d_\infty )$.

The assumption that each horofunction is a Busemann point and that any two distinct Busemann points lie at infinite detour distance from each other is a type of regularity condition on the asymptotic geometry of the space, which is satisfied by numerous metric spaces, such as finite dimensional normed spaces with smooth norms [24], Hilbert geometries on bounded strictly convex domains with $C^1$-boundary [25], and, as we shall see in Lemma 3.3, Kobayashi metric spaces $(D,k_D)$, where $D\subset \mathbb {C}^n$ is a bounded strongly convex domain with $C^3$-boundary.

It turns out that the parts of the horofunction boundary and the detour distance in product metric spaces have a special structure that is closely linked to a quotient space of $(\mathbb {R}^n,2\Vert \cdot \Vert _\infty )$, where $\Vert x\Vert _\infty =\max _j |x_j|$. More precisely, if we let $\mathrm {Sp}(\mathbf {1}) :=\{\lambda (1,\ldots ,1)\in \mathbb {R}^n:\lambda \in \mathbb {R}\}$, then the quotient space $\mathbb {R}^n/\mathrm {Sp}(\mathbf {1})$ with respect to $2\Vert \cdot \Vert _\infty$ has the variation norm as the quotient norm, which is given by

$$\begin{aligned} \Vert \overline{x}\Vert _{\mathrm {var}} :=\max _jx_j + \max _j (-x_j)~\hbox {for }~\overline{x} \in \mathbb {R}^n/\mathrm {Sp}(\mathbf {1}), \end{aligned}$$

(1.1)

see [17, Sect. 4]. It is known, e.g. [14, Proposition 2.2.4], that $(\mathbb {R}^n/\mathrm {Sp}(\mathbf {1}), \Vert \cdot \Vert _{\mathrm {var}})$ is isometric to the Hilbert metric space on the open $(n-1)$-dimensional simplex.

We show in Theorem 2.8 that if, for $j=1,\ldots ,q$, we have that $(N_j,\rho _j)$ is a proper geodesic metric space such that all its horofunctions are Busemann points, and $\delta (h_j,h_j')=\infty$ for all $h_j\ne h_j'$ Busemann points of $(N_j,\rho _j)$, then each part of $(\prod _{j=1}^qN_j, d_\infty )$ is isometric to $(\mathbb {R}^n/\mathrm {Sp}(\mathbf {1}), \Vert \cdot \Vert _{\mathrm {var}})$ for some $1\le n\le q$.

The horofunctions for the product of two metric spaces have been considered by Walsh [26, Sect. 8]. Some of our results are extensions of his work to arbitrary finite products, and the ideas of some of the proofs are quite similar. For the reader’s convenience, we give full proofs and provide comments on the relation with the work in [26] where relevant.

The work in this paper has links to work by Bracci and Gaussier [7] who studied the interaction between topological properties and the metric geometry of hyperbolic complex spaces. It is also worth mentioning that various other aspects of the metric geometry of product metric spaces have been studied in context of Teichmüller space in [9, 20].

2 The Metric Compactification of Product Spaces

In our set-up, we will follow the terminology in [12], which contains further references and background on the metric compactification.

Let (M, d) be a metric space, and let $\mathbb {R}^M$ be the space of all real functions on M equipped with the topology of pointwise convergence. Fix $b\in M$, which is called the basepoint. Let $\mathrm {Lip}^1_b(M)$ denote the set of all functions $h\in \mathbb {R}^M$ such that $h(b)=0$ and h is 1-Lipschitz, i.e. $|h(x)-h(y)|\le d(x,y)$ for all $x,y\in M$. Then $\mathrm {Lip}^1_b(M)$ is a closed subset of $\mathbb {R}^M$. Moreover, as

$$\begin{aligned} |h(x)|= |h(x)-h(b)|\le d(x,b) \end{aligned}$$

for all $h\in \mathrm {Lip}^1_b(M)$ and $x\in M$, we get that $\mathrm {Lip}_b^1(M)\subseteq [-d(x,b),d(x,b)]^M$, which is compact by Tychonoff’s theorem. Thus, $\mathrm {Lip}^1_b(M)$ is a compact subset of $\mathbb {R}^M$.

Now for $y\in M$ consider the real-valued function

$$\begin{aligned} h_{y}(z) := d(z,y)-d(b,y)\,\hbox {with}\> z\in M. \end{aligned}$$

Then $h_y(b)=0$ and $|h_y(z)-h_y(w)| = |d(z,y)-d(w,y)|\le d(z,w)$. Thus, $h_y\in \mathrm {Lip}_b^1(M)$ for all $y\in M$. The closure of $\{h_y:y\in M\}$ is called the metric compactification of M and is denoted $\overline{M}^h$. The boundary $\partial \overline{M}^h:= \overline{M}^h\setminus \{h_y:y\in M\}$ is called the horofunction boundary of M, and its elements are called horofunctions. Given a horofunction h and $r\in \mathbb {R}$, the set $\mathcal {H}(h,r):=\{x\in M:h(x)<r\}$ is a called a horoball.

We will assume that the metric space (M, d) is proper, meaning that all closed balls are compact. Such metric spaces are separable, since every compact metric space is separable. It is known that if (M, d) is separable, then the topology of pointwise convergence on $\mathrm {Lip}^1_b(M)$ is metrizable, and hence each horofunction is the limit of a sequence of functions $(h_{y^n})$ with $y^n\in M$ for all $n\ge 1$. In general, however, horofunctions are limits of nets.

A curve $\gamma :I\rightarrow (M,d)$, where I is a possibly unbounded interval in $\mathbb {R}$, is called a geodesic path if

$$\begin{aligned} d(\gamma (s),\gamma (t))=|s-t|\,\hbox {for all}~s,t\in I. \end{aligned}$$

The metric space (M, d) is said to be a geodesic space if for each $x,y\in M$ there exists a geodesic path $\gamma :[a,b]\rightarrow M$ with $\gamma (a) =x$ and $\gamma (b) =y$. A proof of the following well-known fact can be found in [16, Lemma 2.1].

Lemma 2.1

If (M, d) is a proper geodesic metric space, then $h\in \partial \overline{M}^h$ if and only if there exists a sequence $(y^n)$ in M such that $h_{y^n}\rightarrow h$ and $d(y^n,b)\rightarrow \infty$ as $n\rightarrow \infty$.

It should be noted that in the previous lemma, it is necessary to assume that the metric space is proper. Indeed, consider the star graph with centre vertex b and edges $E_n = \{b,v^n\}$ of length n for $n\in \mathbb {N}$. Then the sequence $(v^n)$ in the resulting path metric space, with basepoint b, satisfies

$$\begin{aligned} \lim _{n\rightarrow \infty } h_{v^n}(x) = \lim _{n\rightarrow \infty }d(x,v^n)-d(b,v^n) = d(x,b) = h_b(x) \end{aligned}$$

for all x, and hence does not yield a horofunction.

A sequence $(y^n)$ in (M, d) is called an almost geodesic sequence if $d(y^n,y^0)\rightarrow \infty$ as $n\rightarrow \infty$, and for each $\varepsilon >0$, there exists $N\ge 0$ such that

$$\begin{aligned} d(y^m,y^k) +d(y^k,y^0) - d(y^m,y^0) <\varepsilon ~\hbox {for all }m\ge k\ge N. \end{aligned}$$

The notion of an almost geodesic sequence goes back to Rieffel [23] and was further developed in [5, 15, 24, 27]. In particular, any almost geodesic sequence yields a horofunction, see [23, Lemma 4.5].

Lemma 2.2

Let (M, d) be a proper geodesic metric space. If $(y^n)$ is an almost geodesic sequence in M, then

$$\begin{aligned} h(x) = \lim _{n\rightarrow \infty } d(x,y^n)-d(b,y^n) \end{aligned}$$

exists for all $x\in M$ and, moreover, $h\in \partial \overline{M}^h$.

Given a proper geodesic metric space (M, d), a horofunction $h\in \overline{M}^h$ is called a Busemann point if there exists an almost geodesic sequence $(y^n)$ in M such that $h(x) = \lim _{n\rightarrow \infty } d(x,y^n)- d(b,y^n)$ for all $x\in M$. We denote the collection of all Busemann points by $\mathcal {B}_M$.

It is known that a product metric space $(\prod _{j=1}^p M_j,d_\infty )$, where

$$\begin{aligned} d_\infty (x,y) = \max _j d_j(x_j,y_j)~\hbox {for}~ x=(x_1,\ldots ,x_p),y=(y_1,\ldots ,y_p)\in \prod _{j=1}^p M_j, \end{aligned}$$

is a proper geodesic metric space, if each $(M_j,d_j)$ is a proper geodesic metric space, see for instance [21, Proposition 2.6.6].

The horofunctions of a product proper geodesic metric spaces have a special form, as the following theorem shows. This theorem is an extension of [26, Proposition 8.1], and the basic idea of the proof is the same.

Theorem 2.3

For $j=1,\ldots ,p$ let $(M_j,d_j)$ be proper geodesic metric spaces. Suppose that h is a horofunction of $(\prod _{j=1}^p M_j,d_\infty )$ with basepoint $b=(b_1,\ldots ,b_p)$. If $(y^n)$ is a sequence in $\prod _{j=1}^p M_j$ converging to h, then there exist $J\subseteq \{1,\ldots ,p\}$ non-empty, horofunctions $h_j$ in $\overline{M_j}^h$ with basepoint $b_j$ for $j\in J$, $\alpha \in \mathbb {R}^J$ with $\min _{j\in J} \alpha _j=0$, and a subsequence $(y^{n_k})$ such that

• (1) $d_\infty (b,y^{n_k}) -d_k(b_j,y^{n_k}_j)\rightarrow \alpha _j$ for all $j\in J$,

• (2) $d_\infty (b,y^{n_k}) -d_k(b_i,y^{n_k}_i)\rightarrow \infty$ for all $i\not \in J$,

• (3) $h_{y^{n_k}_j}\rightarrow h_j$ for all $j\in J$.

Moreover, h is of the form,

$$\begin{aligned} h(x) = \max _{j\in J} h_j(x_j) -\alpha _j ~\hbox {for}~ x=(x_1,\ldots ,x_p)\in \prod _{j=1}^p M_j. \end{aligned}$$

(2.1)

Proof

Let $(y^n)$ be a sequence in $\prod _{j=1}^p M_j$ such that $(h_{y^n})$ converges to a horofunction h. So $h(x)=\lim _{n\rightarrow \infty } d_\infty (x,y^n)-d_\infty (b,y^n)$ for all $x\in \prod _{j=1}^p M_j$. As the product metric space is a proper geodesic metric space, it follows from Lemma 2.1 that $d_\infty (b,y^n)\rightarrow \infty$ as $n\rightarrow \infty$. Write $y^n :=(y_1^n,\ldots ,y^n_p)$ and let $\alpha ^n_j:= d_\infty (b,y^n)-d_j(b_j,y^n_j)\ge 0$ for all $j=1,\ldots ,p$ and $n\ge 0$.

We may assume, after taking a subsequence, that $h_{y^n_j}(\cdot ) := d_j(\cdot , y^n_j)-d_j(b_j,y^n_j)$ converges to $h_j\in \overline{M_j}^h$ and $\alpha ^n_j\rightarrow \alpha _j\in [0,\infty ]$ for all $j\in \{1,\ldots ,p\}$, and $\alpha ^n_{j_0}=0$ for all $n\ge 0$ for some fixed $j_0\in \{1,\ldots ,p\}$. Let $J:=\{j:\alpha _j<\infty \}$ and note that $j_0\in J$. So,

$$\begin{aligned} h(x)= & {} \lim _{n\rightarrow \infty } d_\infty (x,y^n)-d_\infty (b,y^n) = \lim _{n\rightarrow \infty } \max _j (d_j(x_j,y_j^n)-d_j(b_j,y_j^n) - \alpha _j^n)\\&= \max _{j\in J} h_j(x_j)-\alpha _j. \end{aligned}$$

To complete the proof note that $\alpha _j<\infty$ implies that $d_j(b_j,y^n_j)\rightarrow \infty$, and hence by Lemma 2.1, we find that $h_j$ is a horofunction of $(M_j,d_j)$ for $j\in J$. $\square$

The following notion will be useful in the sequel. A path $\gamma :[0,\infty )\rightarrow (M,d)$ is called an almost geodesic ray if $d(\gamma (t),\gamma (0))\rightarrow \infty$, and for each $\varepsilon >0$, there exists $T\ge 0$ such that

$$\begin{aligned} d(\gamma (t),\gamma (s)) +d(\gamma (s),\gamma (0)) - d(\gamma (t),\gamma (0)) <\varepsilon ~\hbox {for all }t\ge s\ge T. \end{aligned}$$

Let $(y^n)$ be an almost geodesic sequence in a geodesic metric space (M, d) and assume that

$$\begin{aligned} d(y^n,y^0)<d(y^{n+1},y^0)~\hbox {for all}~n\ge 0. \end{aligned}$$

(2.2)

For simplicity, we write $\Delta _n:= d(y^{n},y^0)$ and we let $\gamma _n:[0,d(y^{n+1},y^n)]\rightarrow (M,d)$ be a geodesic path connecting $y^n$ and $y^{n+1}$, i.e. $\gamma _n(0) = y^n$ and $\gamma _n(d(y^{n+1},y^n)) = y^{n+1}$. for all $n\ge 0$.

We write $I_n:=[\Delta _n,\Delta _{n+1}]$ and let $\bar{\gamma }_n:I_n\rightarrow (M,d)$ be the affine reparametrisation of $\gamma _n$ given by

$$\begin{aligned} \bar{\gamma }_n(t) := \gamma _n\left( \frac{d(y^{n+1},y^n)}{ \Delta _{n+1} - \Delta _n}(t - \Delta _n)\right) \,\hbox {for all}~ t \in I_n. \end{aligned}$$

We call the path $\bar{\gamma }:[0,\infty )\rightarrow (M,d)$ given by

$$\begin{aligned} \bar{\gamma }(t) :=\bar{\gamma }_n(t)\hbox { for }t\in I_n \end{aligned}$$

a ray induced by $(y^n)$. Note that $\bar{\gamma }$ is well defined for all $t\ge 0$ by (2.2).

Lemma 2.4

If $(y^n)$ is an almost geodesic sequence in a geodesic metric space (M, d) converging to a horofunction h and satisfying (2.2), then each ray, $\bar{\gamma }$, induced by $(y^n)$ satisfies:

• (i) $\bar{\gamma }$ is an almost geodesic ray and $h_{\bar{\gamma }(t)}\rightarrow h$ as $t\rightarrow \infty$,

• (ii) the map $t\mapsto d(\bar{\gamma }(t),\bar{\gamma }(0))$ is continuous on $[0,\infty )$.

Proof

We first show that for each $\varepsilon >0$, there exists $T\ge 0$ such that

$$\begin{aligned} d(\bar{\gamma }(t),y^n) +d(y^n,y^0) - d(\bar{\gamma }(t),y^0)<\varepsilon ~ \hbox {for all} ~t\ge T ~and~ n\ge 0~ with ~t\in I_n.\nonumber \\ \end{aligned}$$

(2.3)

To get this inequality, just note that there exists $N\ge 0$ such that

$$\begin{aligned} d(\bar{\gamma }(t),y^n) +d(y^n,y^0) - d(\bar{\gamma }(t),y^0)= & {} d(y^{n+1}, \bar{\gamma }(t))+d(\bar{\gamma }(t),y^n) +d(y^n,y^0) \\&- d(\bar{\gamma }(t),y^0)-d(y^{n+1}, \bar{\gamma }(t))\\\le & {} d(y^{n+1},y^n) {+}d(y^n,y^0) {-} d(y^{n+1},y^0){<}\varepsilon , \end{aligned}$$

for all $t\in I_n$ and $n\ge N$, as $(y^n)$ is an almost geodesic sequence. So to get (2.3), we can take $T=\Delta _N$.

To prove that $\bar{\gamma }$ is an almost geodesic ray, we need to show that for each $\varepsilon >0$, there exists $S\ge 0$ such that

$$\begin{aligned} d(\bar{\gamma }(t),\bar{\gamma }(s)) +d(\bar{\gamma }(s),\bar{\gamma }(0)) - d(\bar{\gamma }(t),\bar{\gamma }(0)) <\varepsilon ~\hbox {for all }t\ge s\ge S. \end{aligned}$$

Suppose that $t>s$ are such that $t\in I_n$ and $s\in I_k$ with $n>k$. Then by using (2.3), we know that for all n and k large,

$$\begin{aligned} d(\bar{\gamma }(t),\bar{\gamma }(s))+ & {} d(\bar{\gamma }(s),\bar{\gamma }(0)) - d(\bar{\gamma }(t),\bar{\gamma }(0)) \le d(\bar{\gamma }(t),\bar{\gamma }(s)) +d(\bar{\gamma }(s),y^k)+ d(y^k,y^0)\\&\qquad - d(\bar{\gamma }(t),y^0)\\\le & {} d(\bar{\gamma }(t),y^n)+d(y^n,\bar{\gamma }(s)) +d(\bar{\gamma }(s),y^k)\\&\qquad + d(y^k,y^0) - d(\bar{\gamma }(t),y^0)\\< & {} - d(y^n,y^0)+d(y^n,\bar{\gamma }(s)) +d(\bar{\gamma }(s),y^k)\\&\qquad + d(y^k,y^0) +\varepsilon \\\le & {} - d(y^n,y^0)+d(y^n,y^{k+1})+d(y^{k+1},\bar{\gamma }(s))\\&\qquad +d(\bar{\gamma }(s),y^k) + d(y^k,y^0) +\varepsilon \\= & {} - d(y^n,y^0)+d(y^n,y^{k+1})+d(y^{k+1},y^k)\\&\qquad + d(y^k,y^0) +\varepsilon \\< & {} - d(y^n,y^0)+d(y^n,y^{k+1})\\&\qquad + d(y^{k+1},y^0) +2\varepsilon <3\varepsilon .\\ \end{aligned}$$

Finally suppose that $t\ge s$ are such that $t,s\in I_n$. Then for all $n\ge 0$ large we have that

$$\begin{aligned} d(\bar{\gamma }(t),\bar{\gamma }(s)) +d(\bar{\gamma }(s),\bar{\gamma }(0)) - d(\bar{\gamma }(t),\bar{\gamma }(0))= & {} d(\bar{\gamma }(t),y^n) -d(y^n,\bar{\gamma }(s))+d(\bar{\gamma }(s),\bar{\gamma }(0)) \\&\qquad - d(\bar{\gamma }(t),\bar{\gamma }(0)) \\\le & {} d(\bar{\gamma }(t),y^n)+ d(y^n,y^0) - d(\bar{\gamma }(t),y^0)< \varepsilon . \end{aligned}$$

As $\bar{\gamma }$ is an almost geodesic ray, we know by [23, Lemma 4.5] that $h_{\bar{\gamma }}(t)\rightarrow h'$, where $h'$ is a horofunction. As $\bar{\gamma }(\Delta _n)=y^n$ for all n, we get that $h'=h$.

To prove the second assertion, we note that the affine map

$$\begin{aligned} t\mapsto \frac{d(y^{n+1},y^n)}{ \Delta _{n+1} - \Delta _n}(t - \Delta _n) \end{aligned}$$

is a continuous map from $I_n$ onto $[0,d(y^{n_1},y^n)]$, and the map $\gamma _n:[0,d(y^{n+1},y^n)]\rightarrow (M,d)$ is continuous, as $\gamma _n$ is a geodesic. Thus, the map $t\mapsto d(\bar{\gamma }(t),\bar{\gamma }(0))$ is continuous on the interior of the interval $I_n$ for all $n\ge 0$. To get continuity at the endpoints, we simply note that for all $n\ge 0$,

$$\begin{aligned} \lim _{t\rightarrow \Delta _n^-} d(\bar{\gamma }(t), \bar{\gamma }(0)) = d(y^n,\bar{\gamma }(0)) = \lim _{t\rightarrow \Delta _n^+} d(\bar{\gamma }(t), \bar{\gamma }(0)), \end{aligned}$$

which completes the proof. $\square$

Lemma 2.5

If $(y^n)$ is an almost geodesic sequence in a geodesic metric space (M, d) satisfying (2.2) and $\bar{\gamma }$ is a ray induced by $(y^n)$, then for each sequence $(\beta ^n)$ in $[0,\infty )$ with $\beta ^{n+1}>\beta ^n$ for all $n\ge 0$, there exists sequence $(t^n)$ in $[0,\infty )$ with $t^{n+1}>t^n$ for all $n\ge 0$ such that $d(\bar{\gamma }(t^n),\bar{\gamma }(0)) = \beta ^n$ for all $n\ge 0$.

Proof

Note that as $\overline{\gamma }:[0,\infty )\rightarrow M$ is an almost geodesic by Lemma 2.4(i), we know that $d(\overline{\gamma }(t),\overline{\gamma }(0))\rightarrow \infty$ as $t\rightarrow \infty$. From Lemma 2.4(ii) we know that $\overline{\gamma }$ is continuous on $[0,\infty )$, so there exists $t_0^n\ge 0$ such that $d(\overline{\gamma }(t_0^n),\overline{\gamma }(0)) =\beta ^n$. Now if we let $t^n:=\inf \{t\ge 0:d(\overline{\gamma }(t),\overline{\gamma }(0)) =\beta ^n\}$, then by continuity of $\overline{\gamma }$ we have that $d(\overline{\gamma }(t^n),\overline{\gamma }(0)) =\beta ^n$ and $t^{n+1}>t^n$ for all $n\ge 0$. $\square$

2.1 Detour Distance

Suppose that (M, d) is a proper geodesic metric space. Given two Busemann points $h_1,h_2\in \partial \overline{M}^h$, the detour cost is given by

$$\begin{aligned} H(h_1,h_2) := \inf _{(z^n)}\lim \inf _n d(b,z^n) +h_2(z^n), \end{aligned}$$

(2.4)

where the infimum is taken over all sequences $(z^n)$ such that $h_{z^n}$ converges to $h_1$. It is known, see [15, Lemma 3.1], that if $(z^n)$ is an almost geodesic sequence converging to $h_1$ and $(w^m)$ converges to $h_2$, then

$$\begin{aligned} H(h_1,h_2) = \lim _{n\rightarrow \infty }\left( d(b,z^n) +\lim _{m\rightarrow \infty } d(z^n,w^m) -d(b,w^m)\right) = \lim _{n\rightarrow \infty } d(b,z^n) +h_2(z^n). \end{aligned}$$

The detour distance is given by

$$\begin{aligned} \delta (h_1,h_2) := H(h_1,h_2)+H(h_2,h_1). \end{aligned}$$

Note that for all $m,n\ge 0$, we have that

$$\begin{aligned} d(b,z^n) + d(z^n,w^m) -d(b,w^m)\ge 0, \end{aligned}$$

so that $H(h_1,h_2)\ge 0$ for all $h_1,h_2\in \partial \overline{M}^h$. It is, however, possible for $H(h_1,h_2)$ to be infinite. It can be shown, see [15, Sect. 3] or [27, Sect. 2], that the detour distance is independent of the basepoint.

The detour distance was introduced in [5] and has been exploited and further developed in [15, 27]. It is known, see for instance [15, Sect. 3] or [27, Sect. 2], that on $\mathcal {B}_M\subseteq \partial \overline{M}^h$, the detour distance is symmetric, satisfies the triangle inequality, and $\delta (h_1,h_2) =0$ if and only if $h_1=h_2$. This yields a partition of $\mathcal {B}_M$ into equivalence classes, where $h_1$ and $h_2$ are said to be equivalent if $\delta (h_1,h_2) <\infty$. The equivalence class of h will be denoted by $\mathcal {P}(h)$. Thus, the set of Busemann points, $\mathcal {B}_M$, is the disjoint union of metric spaces under the detour distance, which are called parts of $\mathcal {B}_M$.

Isometric embeddings between proper geodesic metric spaces can be extended to the parts of the metric spaces as detour distance isometries. Indeed, suppose that $\varphi :(M,d)\rightarrow (N,\rho )$ is an isometric embedding, i.e. $\rho (\varphi (x),\varphi (y)) =d(x,y)$ for all $x,y\in M$. (Note that $\varphi$ need not be onto.) If h is a Busemann point of (M, d) with basepoint b and $(z^n)$ is an almost geodesic sequence such that $(h_{z^n})$ converges to h, then $(u^n)$, with $u^n:=\varphi (z^n)$ for $n\ge 0$, is an almost geodesic sequence in $(N,\rho )$, and hence $(h_{u^n})$ converges to a Busemann point, say $\varphi (h)$, of $(N,\rho )$ with basepoint $\varphi (b)$.

We note that $\varphi (h)$ is independent of the almost geodesic sequence $(z^n)$. To see this, let $(w^n)$ be another almost geodesic such that $(h_{w^n})$ converges to h. Write $v^n:=\varphi (w^n)$ for $n\ge 0$ and let $\varphi (h)'$ be the limit of $(h_{v^n})$. Then

$$\begin{aligned} H(h,h)= & {} \lim _{n\rightarrow \infty } d(w^n,b) + \lim _{m\rightarrow \infty } d(w^n,z^m) -d(b,z^m)\\= & {} \lim _{n\rightarrow \infty } \rho (v^n,\varphi (b)) + \lim _{m\rightarrow \infty } \rho (v^n,u^m) -\rho (\varphi (b),u^m)\\= & {} H(\varphi (h)',\varphi (h)). \end{aligned}$$

Likewise, $H(\varphi (h),\varphi (h)') = H(h,h)$, and we deduce that $\delta (\varphi (h)',\varphi (h))= H(\varphi (h)',\varphi (h))+H(\varphi (h),\varphi (h)')=\delta (h,h) =0$, which shows that $\varphi (h)'=\varphi (h)$, as $\varphi (h)'$ and $\varphi (h)$ are Busemann points. Thus, there exists a well-defined map $\Phi :\mathcal {B}_M\rightarrow \mathcal {B}_N$ given by $\Phi (h) :=\varphi (h)$.

Lemma 2.6

If $\varphi :(M,d)\rightarrow (N,\rho )$ is an isometric embedding, then $\Phi (\mathcal {P}(h))\subseteq \mathcal {P}(\varphi (h))$  for all Busemann points h of (M, d) and

$$\begin{aligned} \delta (h',h) = \delta (\Phi (h'),\Phi (h))\hbox { for all } h,h'\in \mathcal {B}_M. \end{aligned}$$

Proof

Let $(z^n)$ and $(w^n)$ be almost geodesic sequences such that $(h_{z^n})$ converges to h and $(h_{w^n})$ converges to $h'$ in (M, d) with basepoint b. Then

$$\begin{aligned} H(h',h)= & {} \lim _{n\rightarrow \infty } d(w^n,b) + \lim _{m\rightarrow \infty } d(w^n,z^m) -d(b,z^m)\\= & {} \lim _{n\rightarrow \infty } \rho (v^n,\varphi (b)) + \lim _{m\rightarrow \infty } \rho (v^n,u^m) -\rho (\varphi (b),u^m)\\= & {} H(\varphi (h)',\varphi (h)). \end{aligned}$$

Likewise, $H(h,h') = H(\varphi (h),\varphi (h)')$, so that $\delta (h',h) = \delta (\Phi (h'),\Phi (h))$, which completes the proof. $\square$

It could happen that all parts consist of a single Busemann point, but there are also natural instances where there are nontrivial parts. In case of products of metric spaces coming from proper geodesic metric spaces, it turns out that the parts and the detour distance have a special structure that is linked to the quotient space, $(\mathbb {R}^n/\mathrm {Sp}(\mathbf {1}), \Vert \cdot \Vert _{\mathrm {var}})$ given in (1.1), as shown by the following proposition.

Proposition 2.7

If, for $j=1,\ldots , p$, $(M_j,d_j)$ is a proper geodesic metric space with almost geodesic sequence $(y^n_j)$ and corresponding Busemann point $h_j$ with basepoint $y^0_j$, and $J\subseteq \{1,\ldots ,p\}$ is non-empty, then the following assertions hold:

• (i) For $\alpha \in \mathbb {R}^J$ with $\min _{j\in J}\alpha _j=0$ the function $h:(\prod _{j=1}^p M_j,d_\infty )\rightarrow \mathbb {R}$ given by,

  $$\begin{aligned} h(x) =\max _{j\in J} h_j(x_j)-\alpha _j,\hbox { for }x\in \prod _{j=1}^p M_j, \end{aligned}$$

  (2.5)

  is a Busemann point with basepoint $y^0=(y^0_1,\ldots ,y^0_p)$. Moreover, there exists an almost geodesic sequence $(z^n)$ converging to h, where $(z^n_j)$ is an almost geodesic converging to $h_j$ for $j\in J$ such that for all $n\ge 1$, we have that $d_\infty (z^n,y^0) -d_j(z^n_j,y^0_j) =\alpha _j$ for $j\in J$, and $d_i(z^n_i,y^0_i) =0$ for all $i\not \in J$.

• (ii) If $\beta \in \mathbb {R}^J$ with $\min _{j\in J}\beta _j=0$ and $h'$ is a Busemann point with basepoint $y^0$ of the form,

  $$\begin{aligned} h'(x) =\max _{j\in J} h_j(x_j)-\beta _j,\hbox { for}~ x\in \prod _{j=1}^p M_j, \end{aligned}$$

  then $\delta (h,h') = \Vert \alpha -\beta \Vert _{\mathrm {var}}$.

• (iii) For h as in (2.5) the part $(\mathcal {P}(h),\delta )$ contains an isometric copy of $(\mathbb {R}^J/\mathrm {Sp}(\mathbf {1}), \Vert \cdot \Vert _{\mathrm {var}})$.

Proof

We know there exists an almost geodesic sequence $(y^n_j)$ in $(M_j,d_j)$ such that $h_{y^n_j}\rightarrow h_j$ for each $j\in J$. As $d_j(y^n_j,b_j)\rightarrow \infty$, we can take a subsequence and assume that $d_j(y^{n+1}_j,y^0_j)>d_j(y^n_j,y^0_j)>\alpha _j$ for all $n\ge 1$. Let $\bar{\gamma }_j$ be a ray induced by $(y^n_j)$.

For $j\in J$ we get from Lemma 2.5 a sequence $(t^n_j)$ in $[0,\infty )$ with $t^0_j=0$ and

$$\begin{aligned} d_j(\bar{\gamma }_j(t^n_j),y^0_j) = (\max _{i\in J}d_i(y^n_i,y^0_i))-\alpha _j\ge 0\hbox { for all }n\ge 1. \end{aligned}$$

Let $z^0:=(y^0_1,\ldots ,y^0_p)$ and for $n\ge 1$ define $z^n=(z^n_1,\ldots ,z^n_p)\in \prod _{j=1}^p M_j$ by $z^n_j:=\bar{\gamma }_j(t^n_j)$ if $j\in J$, and $z^n_j :=y^0_j$ otherwise.

As $\min _{j\in J}\alpha _j=0$, we get by construction that

$$\begin{aligned} d_\infty (z^n,z^0) = \max _{i\in J}d_i(y^n_i,y^0_i) = d_j(z^n_j,z^0_j) +\alpha _j\hbox { for all }n\ge 1\hbox { and } j\in J. \end{aligned}$$

Moreover, it follows from Lemma 2.4 that $(z^n_j)$ is an almost geodesic converging to $h_j$ for $j\in J$.

We claim that $(z^n)$ is an almost geodesic sequence in $(\prod _{j=1}^p M_j,d_\infty )$. Indeed, note that for $n\ge k\ge 0$, we have that

$$\begin{aligned} d_\infty (z^n,z^k) +d_\infty (z^k,z^0) - d_\infty (z^n,z^0) = d_j(z^n_j,z^k_j) + d_\infty (z^k,z^0) - d_\infty (z^n,z^0) \end{aligned}$$

for some $j=j(n,k)\in J$, as $d_j(z^n_j,z_j^k)=0$ for all $j\not \in J$. As J is non-empty, we find for all $n\ge k$ large that

$$\begin{aligned} d_\infty (z^n,z^k) +d_\infty (z^k,z^0) - d_\infty (z^n,z^0) = d_j(z^n_j,z^k_j) + d_j(z^k_j,z^0_j) +\alpha _j- d_j(z^n_j,z^0_j) -\alpha _j< \varepsilon . \end{aligned}$$

Also for $n\ge 0$ large and $x\in \prod _{j=1}^p M_j$, we have that

$$\begin{aligned} h_{z^n}(x) = \max _{j\in J}( d_j(x_j,z^n_j) - d_\infty (z^n,z^0)) = \max _{j\in J}( d_j(x_j,z^n_j) - d_j(z^n_j,z^0_j) -\alpha _j). \end{aligned}$$

Letting $n\rightarrow \infty$ gives

$$\begin{aligned} h(x) = \max _{j\in J}h_j(x_j) -\alpha _j\hbox { for all}~ x\in \prod _{j=1}^p M_j \end{aligned}$$

and shows that h is a Busemann point with basepoint $y^0=(y^0_1,\ldots ,y^0_p)$. This completes the proof of assertion (i).

To prove the second assertion, we know from the first assertion that there exists an almost geodesic sequence $(w^n)$ converging to $h'$, where $(w^n_j)$ is an almost geodesic converging to $h_j$ and $d_\infty (w^n,y^0)-d_j(w_j^n,y^0_j)=\beta _j$ for $j\in J$. So, we get that

$$\begin{aligned} \max _{j\in J}\,(\beta _j-\alpha _j)= & {} \max _{j\in J}\,(H(h_j,h_j) +\beta _j-\alpha _j ) \\= & {} \max _{j\in J}\,( \lim _{n\rightarrow \infty } d_j(w^n_j,y^0_j) +\beta _j+h_j(w^n_j) -\alpha _j)\\= & {} \max _{j\in J}\,(\lim _{n\rightarrow \infty } d_\infty (w^n,y^0) +h_j(w^n_j) -\alpha _j)\\= & {} \lim _{n\rightarrow \infty }\max _{j\in J}(d_\infty (w^n,y^0) + h_j(w^n_j) -\alpha _j)\\= & {} \lim _{n\rightarrow \infty } d_\infty (w^n,y^0) +h(w^n). \end{aligned}$$

Interchanging the roles of h and $h'$, we find that

$$\begin{aligned} \delta (h',h) = H(h',h) + H(h,h') = \max _{j\in J}(\beta _j-\alpha _j)+ \max _{j\in J} (\alpha _j-\beta _j) =\Vert \alpha -\beta \Vert _{\mathrm {var}}. \end{aligned}$$

The final assertion is a direct consequence of the previous two, as $(S,\Vert \cdot \Vert _{\mathrm {var}})$ with $S:=\{\alpha \in \mathbb {R}^J:\min _{j\in J}\alpha _j =0\}$ is isometric to $(\mathbb {R}^J/\mathrm {Sp}(\mathbf {1}), \Vert \cdot \Vert _{\mathrm {var}})$. $\square$

Proposition 2.7 is related to [26, Propositions 8.3 and 8.4], where the Busemann points for the product of two metric spaces are characterised and the detour cost is determined.

It is interesting to understand when a part $(\mathcal {P}(h),\delta )$ is isometric to $(\mathbb {R}^J/\mathrm {Sp}(\mathbf {1}), \Vert \cdot \Vert _{\mathrm {var}})$.

Theorem 2.8

If, for $j=1,\ldots ,q$, $(N_j,\rho _j)$ is a proper geodesic metric space such that all horofunctions are Busemann points, and $\delta (h_j,h_j')=\infty$ for all $h_j\ne h_j'$ Busemann points of $(N_j,\rho _j)$, then every horofunction h of $(\prod _{j=1}^q N_j,d_\infty )$ is a Busemann point, and $(\mathcal {P}(h),\delta )$ is isometric to $(\mathbb {R}^{J}/\mathrm {Sp}(\mathbf {1}),\Vert \cdot \Vert _\mathrm {var})$ for some $J\subseteq \{1,\ldots ,q\}$.

Proof

Let h be a horofunction of $(\prod _{j=1}^q N_j,d_\infty )$ with respect to basepoint $b=(b_1,\ldots ,b_q)$. By Theorem 2.3 we know that h is of the form

$$\begin{aligned} h(x) = \max _{j\in J} h_j(x_j) -\alpha _j\hbox { for} ~x\in \prod _{j=1}^q N_j, \end{aligned}$$

and $h_j$ is a horofunction of $(N_j,\rho _j)$ with respect to basepoint $b_j$ for each $j\in J$. As each horofunction of $(N_j,\rho _j)$, is a Busemann point, there exists an almost geodesic sequence $(y^n_j)$ such that $(h_{y^n_j})$ converges to $h_j$ with basepoint $b_j$.

For $j\not \in J$ let $y^0_j=b_j$ and define $y^0:=(y^0_1,\ldots ,y^0_q)$. Let $h^*_{j}$ be the Busemann point obtained by changing the basepoint of $h_j$ to $y^0_j$, so $h^*_{j}(x_j) := h_{j}(x_j) - h_{j}(y^0_j)$. Now note that if we change the basepoint for h to $y^0$, we get the Busemann point

$$\begin{aligned} h^*(x):= & {} h(x) -h(y^0)\\= & {} \max _{j\in J} (h_j(x_j) -\alpha _j) - \max _{i\in J}\,( h_i(y^0_i) -\alpha _i)\\= & {} \max _{j\in J}\,( h^*_{j}(x_j) +h_{j}(y^0_j) -\alpha _j - \max _{i\in J}\, (h_i(y^0_i) -\alpha _i))\\= & {} \max _{j\in J} h^*_{j}(x_j) -\gamma _j,\\ \end{aligned}$$

where $\gamma _j : = \max _{i\in J} (h_i(y^0_i) -\alpha _i) -(h_{j}(y^0_j) -\alpha _j) \ge 0$ for $j\in J$ and $\min _{j\in J}\gamma _j =0$. It now follows from Proposition 2.7(i) that $h^*$ is a Busemann point of $(\prod _{j=1}^q N_j,d_\infty )$ with respect to basepoint $y^0$, and hence h is a Busemann point $(\prod _{j=1}^q N_j,d_\infty )$ with respect to basepoint b. Moreover, there exists an almost geodesic sequence $(z^m)$ converging to $h^*$, where $(z^m_j)$ is an almost geodesic converging to $h^*_j$ for $j\in J$, and for all $m\ge 1$ we have that $d_\infty (z^m,y^0) -d_j(z^m_j,y^0_j) =\gamma _j$ for $j\in J$, and $d_i(z^m_i,y^0_i) =0$ for all $i\not \in J$.

To prove the second assertion we note that $(\mathcal {P}(h),\delta )$ is isometric to $(\mathcal {P}(h^*),\delta )$, since $\delta$ is independent of the basepoint. Let $h'$ is a Busemann point of $(\prod _{j=1}^q N_j,d_\infty )$ with respect to basepoint $y^0$ and $(w^n)$ be an almost geodesic converging to $h'$. Then by Theorem 2.3, we know $h'$ is of the form

$$\begin{aligned} h'(x) =\max _{j\in J'} h'_j(x_j)-\beta _j,\hbox { for}~ x\in \prod _{j=1}^q N_j, \end{aligned}$$

(2.6)

and, after taking a subsequence, we may assume that $d_\infty (w^{n}_j,y^0) -d_k(w^{n}_j,y^0_j)\rightarrow \beta _j$ for all $j\in J'$, $d_\infty (w^{n},y^0) -d_i(w^{n}_i,y^0_i)\rightarrow \infty$ for all $i\not \in J'$, and $h_{w^{n}_j}\rightarrow h'_j\in \partial \overline{N_j}^h$ for all $j\in J'$.

We claim that if $J\ne J'$, or, $J=J'$ and $h_k\ne h_k'$ for some $k\in J$, then $\delta (h^*,h')=\infty$. Suppose that $J\ne J'$ and $k\in J$, but $k\not \in J'$. Then

$$\begin{aligned} \lim _{m\rightarrow \infty } d_\infty (w^n,z^m)-d_\infty (y^0,z^m)= & {} \lim _{m\rightarrow \infty } d_\infty (w^n,z^m)-d_k(y^0_k,z^m_k) -\gamma _k\\\ge & {} \lim _{m\rightarrow \infty } d_k(w^n_k,z^m_k)-d_k(y^0_k,z^m_k) -\gamma _k\\\ge & {} -d_k(w^n_k,y^0_k) -\gamma _k,\ \end{aligned}$$

so that

$$\begin{aligned} \lim _{n\rightarrow \infty } \left( d_\infty (w^n,y^0) +\lim _{m\rightarrow \infty } d_\infty (w^n,z^m)-d_\infty (y^0,z^m)\right) \ge \lim _{n\rightarrow \infty } d_\infty (w^n,y^0)-d_k(w^n_k,y^0_k) -\gamma _k =\infty . \end{aligned}$$

Thus, $H(h',h^*)=\infty$ and hence $\delta (h^*,h')=\infty$. The case where $k\in J'$ and $k\not \in J$ can be shown in the same way.

Now suppose that $h^*_k\ne h_k'$ for some $k\in J\cap J'$. By assumption we know that $\delta (h^*_k,h'_k) =\infty$. Note that

$$\begin{aligned} \lim _{n\rightarrow \infty } d_\infty (w^n,y^0) +h^*(w^n)= & {} \lim _{n\rightarrow \infty } d_\infty (w^n,y^0)+\max _{j\in J} h^*_j(w^n_j) -\gamma _j \\\ge & {} \liminf _{n\rightarrow \infty } d_k(w^n_k,y^0_k)+ h^*_k(w^n_k) -\gamma _k.\\ \end{aligned}$$

It now follows from (2.4) that $H(h',h^*) \ge H(h_k',h^*_k)-\gamma _k$. Interchanging the roles of $h^*$ and $h'$, we also get that $H(h^*,h') \ge H(h^*_k,h_k')-\beta _k$, and hence $\delta (h^*,h')\ge \delta (h^*_k,h'_k) -(\gamma _k+\beta _k) =\infty$.

On the other hand, if $J=J'$ and $h^*_j=h_j'$ for all $j\in J$, then it follows from Proposition 2.7(ii) that $\delta (h^*,h') = \Vert \alpha -\beta \Vert _{\mathrm {var}}$. Moreover, it follows from that Proposition 2.7(i) that for each $\beta \in \mathbb {R}^J$ with $\min _{j\in J}\beta _j=0$, there exists a Busemann point in the part of $h^*$ of the form (2.6), and hence $\mathcal {P}(h^*)$ consists of all $h'$ of the form (2.6), where $\min _{j\in J}\beta _j =0$. So if we let $S:=\{\beta \in \mathbb {R}^J:\min _{j\in J}\beta _j =0\}$, then $(\mathcal {P}(h^*),\delta )$ is isometric to $(S,\Vert \cdot \Vert _{\mathrm {var}})$, which in turn is isometric to the quotient space $(\mathbb {R}^{J}/\mathrm {Sp}(\mathbf {1}),\Vert \cdot \Vert _\mathrm {var})$. $\square$

An elementary example is the product space $(\mathbb {R}^n,d_\infty )$ where $d_\infty (x,y) =\max _j |x_j-y_j|$. It is easy to verify that $(\mathbb {R},|\cdot |)$ with basepoint 0 has only two horofunctions, namely $h_+:x\mapsto x$ and $h_-:x\mapsto -x$, both of which are Busemann points and $\delta (h_+,h_-)=\infty$. So, in this case, we see that the horofunctions h of $(\mathbb {R}^n,d_\infty )$ are all Busemann points and of the form,

$$\begin{aligned} h(x) = \max _{j\in J} \pm x_j -\alpha _j, \end{aligned}$$

for some $J\subseteq \{1,\ldots ,n\}$ non-empty and $\alpha \in \mathbb {R}^J$ with $\min _{j\in J}\alpha _j =0$, where the sign is fixed for each $j\in J$, see also [10, Theorem 5.2]. Moreover, $(\mathcal {P}(h),\delta )$ is isometric to $(\mathbb {R}^J/\mathrm {Sp}(\mathbf {1}),\Vert \cdot \Vert _\mathrm {var})$.

We are now in position to prove Theorem 1.2.

Proof of Theorem 1.2

As each $(M_j,d_j)$ contains an almost geodesic sequence $(y^n_j)$, it has a Busemann point, say $h_j$. We know from Proposition 2.7(i) that the function h of the form, $h(x) = \max _{j=1,\ldots ,p} h_j(x_j)$, is a Busemann point of $(\prod _{j=1}^p M_j,d_\infty )$. Moreover, it follows from the third part of the same proposition that $(\mathcal {P}(h),\delta )$ contains an isometric copy of $(\mathbb {R}^p/\mathrm {Sp}(\mathbf {1}),\Vert \cdot \Vert _\mathrm {var})$.

Now suppose, for the sake of contradiction, that there exists an isometric embedding $\varphi :(\prod _{j=1}^p M_j,d_\infty )\rightarrow (\prod _{j=1}^q N_j,d_\infty )$. Then it follows from Lemma 2.6 that the restriction of $\Phi$ to $\mathcal {P}(h)$ yields an isometric embedding of $(\mathcal {P}(h),\delta )$ into $(\mathcal {P}(\Phi (h)),\delta ')$, where $\delta '$ is the detour distance on $\mathcal {P}(\Phi (h))$. It now follows from Theorem 2.8 that $(\mathcal {P}(\Phi (h)),\delta ')$ is isometric to $(\mathbb {R}^n/\mathrm {Sp}(\mathbf {1}),\Vert \cdot \Vert _\mathrm {var})$ for some $n\in \{1,\ldots ,q\}$. So, $\Phi$ yields an isometric embedding of $(\mathbb {R}^p/\mathrm {Sp}(\mathbf {1}),\Vert \cdot \Vert _\mathrm {var})$ into $(\mathbb {R}^n/\mathrm {Sp}(\mathbf {1}),\Vert \cdot \Vert _\mathrm {var})$ with $n<p$, which contradicts Brouwer’s invariance of domains theorem [8]. $\square$

3 Product Domains in $\mathbb {C}^n$

Before we show how we can use Theorem 1.2 to derive Theorem 1.1, we first recall some basic facts concerning the Kobayashi distance, see [13, Chapter 4] for more details. On the disc, $\Delta :=\{z\in \mathbb {C}:|z|<1\}$, the hyperbolic distance is given by

$$\begin{aligned} \rho (z,w) := \log \frac{ 1+\left| \frac{w-z}{1-\bar{z}w}\right| }{1-\left| \frac{w-z}{1-\bar{z}w}\right| }{=}2\tanh ^{-1}\left( 1 -\frac{(1-|w|^2)(1-|z|^2)}{|1-w\bar{z}|^2}\right) ^{1/2} \hbox { for}~z,w{\in }\Delta . \end{aligned}$$

Given a convex domain $D\subseteq \mathbb {C}^n$, the Kobayashi distance is given by

$$\begin{aligned} k_D(z,w) :=\inf \{ \rho (\zeta ,\eta ):\exists f:\Delta \rightarrow D~ \hbox {holomorphic with} f(\zeta )=z ~and~ f(\eta )=w\} \end{aligned}$$

for all $z,w\in D$. It was shown by Lempert [18] that on bounded convex domains, the Kobayashi distance coincides with the Caratheodory distance, which is given by

$$\begin{aligned} c_D(z,w) := \sup _f \,\rho (f(z),f(w)) \hbox { for all}~ z,w\in D, \end{aligned}$$

where the $\sup$ is taken over all holomorphic maps $f:D\rightarrow \Delta$.

It is known, see [2, Proposition 2.3.10], that if $D\subset \mathbb {C}^n$ is bounded convex domain, then $(D,k_D)$ is a proper metric space, whose topology coincides with the usual topology on $\mathbb {C}^n$. Moreover, $(D,k_D)$ is a geodesic metric space containing geodesics rays, see [2, Theorem 2.6.19] or [13, Theorem 4.8.6].

In the case of the Euclidean ball $B^n:=\{(z_1,\ldots ,z_n)\in \mathbb {C}^n:\Vert z\Vert ^2<1\}$, where $\Vert z\Vert ^2 = \sum _i |z_i|^2$, the Kobayashi distance has an explicit formula:

$$\begin{aligned} k_{B^n}(z,w) = 2\tanh ^{-1}\left( 1 -\frac{(1-\Vert w\Vert ^2)(1-\Vert z\Vert ^2)}{|1-\langle z,w\rangle |^2}\right) ^{1/2} \end{aligned}$$

for all $z,w\in B^n$, see [2, Chapters 2.2 and 2.3].

On the other hand, on the polydisc $\Delta ^n:=\{(z_1,\ldots ,z_n)\in \mathbb {C}^n:\max _i |z_i|<1\}$, the Kobayashi distance satisfies

$$\begin{aligned} k_{\Delta ^n}(z,w) =\max _i \rho (z_i,w_i)~\hbox {for all}~ w=(w_1,\ldots ,w_n), z=(z_1,\ldots ,z_n)\in \Delta ^n, \end{aligned}$$

by the product property, see [13, Theorem 3.1.9].

To determine the horofunctions of $(B^n,k_{B^n})$, with basepoint $b=0$, it suffices to consider limits of sequences $(h_{w_n})$, where $w_n\rightarrow \xi \in \partial B^n$ in norm. As

$$\begin{aligned} k_{B^n}(z,w_n) =\log \frac{\left( |1 -\langle z,w_n\rangle | + ( |1 -\langle z,w_n\rangle |^2 - (1-\Vert z\Vert ^2)(1-\Vert w_n\Vert ^2))^{1/2}\right) ^2}{(1-\Vert z\Vert ^2)(1-\Vert w_n\Vert ^2)}, \end{aligned}$$

and

$$\begin{aligned} k_{B^n}(0,w_n) = \log \frac{(1+\Vert w_n\Vert )^2}{1-\Vert w_n\Vert ^2}, \end{aligned}$$

it follows that

$$\begin{aligned} h(z)= & {} \lim _{n\rightarrow \infty } k_{B^n}(z,w_n)-k_{B^n}(0,w_n)\\= & {} \log \frac{( |1 -\langle z,\xi \rangle | +|1 -\langle z,\xi \rangle |)^2}{(1-\Vert z\Vert ^2)(1+\Vert \xi \Vert )^2} \\= & {} \log \frac{ |1 -\langle z,\xi \rangle |^2}{1-\Vert z\Vert ^2}. \end{aligned}$$

for all $z\in B^n$. Thus, if we write

$$\begin{aligned} h_\xi (z) := \log \frac{ |1 -\langle z,\xi \rangle |^2}{1-\Vert z\Vert ^2}~\hbox {for all}~ z\in B^n, \end{aligned}$$

(3.1)

then we obtain $\partial \overline{B^n}^h =\{h_\xi :\xi \in \partial B^n\}$, see also [3, Lemma 2.28] and [11, Remark 3.1]. Moreover, each $h_\xi$ is a Busemann point, as it is the limit induced by the geodesic ray $t\mapsto \frac{e^t-1}{e^t+1}\xi$, for $0\le t<\infty$.

Corollary 3.1

If $h_\xi$ and $h_\eta$ are distinct horofunctions of $(B^n,k_{B^n})$, then $\delta (h_\xi ,h_\eta )=\infty$.

Proof

If $\xi \ne \eta$ in $\partial B^n$, then

$$\begin{aligned} \lim _{z\rightarrow \eta } k_{B^n}(z,0) + h_\xi (z) = \lim _{z\rightarrow \eta } \log \frac{1+\Vert z\Vert }{1-\Vert z\Vert } + \log \frac{ |1 -\langle z,\xi \rangle |^2}{1-\Vert z\Vert ^2} =\infty , \end{aligned}$$

which implies that $\delta (h_\xi ,h_\eta )=\infty$. $\square$

Note that if $n=1$, we recover the well-known expression for the horofunctions of the hyperbolic distance on $\Delta$:

$$\begin{aligned} h_\xi (z) = \log \frac{ |1 -z\overline{\xi }|^2}{1-|z|^2}= \log \frac{ |\xi -z|^2}{1-|z|^2}\hbox { for all}~ z\in \Delta . \end{aligned}$$

Combining (3.1) with Theorem 2.3 and Proposition  2.7, we get the following.

Corollary 3.2

For $B^{n_1}\times \cdots \times B^{n_q}$, the Kobayashi distance horofunctions with basepoint $b=0$ are precisely the functions of the form,

$$\begin{aligned} h(z) = \max _{j\in J} \left( \log \frac{ |1 -\langle z_j,\xi _j\rangle |^2}{1-\Vert z_j\Vert ^2}-\alpha _j\right) , \end{aligned}$$

where $J\subseteq \{1,\ldots ,q\}$ non-empty, $\xi _j\in \partial B^{n_j}$ for $j\in J$, and $\min _{j\in J} \alpha _{j}=0$. Moreover, each horofunction is a Busemann point, and $(\mathcal {P}(h),\delta )$ is isometric to $(\mathbb {R}^J/\mathrm {Sp}(\mathbf {1}),\Vert \cdot \Vert _\mathrm {var})$.

Corollary 3.2 should be compared with [2, Proposition 2.4.12].

A similar result holds for more general product domains. We know from [2, Theorem 2.6.45] that for each $\xi \in \partial D$, there exists a unique geodesic ray $\gamma _\xi :[0,\infty )\rightarrow D$ such that $\gamma _\xi (0)= b$ and $\lim _{t\rightarrow \infty }\gamma _\xi (t) =\xi$, if $D\subset \mathbb {C}$ is bounded strongly convex domain with $C^3$-boundary. We will denote the corresponding Busemann point by $h_\xi :D\rightarrow \mathbb {R}$, so

$$\begin{aligned} h_\xi (z) =\lim _{t\rightarrow \infty } k_D(z,\gamma _\xi (t)) -k_D(b,\gamma _\xi (t)). \end{aligned}$$

Lemma 3.3

If $D\subset \mathbb {C}^n$ is a bounded strongly convex domain with $C^3$-boundary, then each horofunction of $(D,k_D)$ is a Busemann point and of the form $h_\xi$ for some $\xi \in \partial D$. Moreover, $\delta (h_\xi ,h_\eta ) =\infty$ if $\xi \ne \eta$. If $D = \prod _{i=1}^r D_i$, where each $D_i$ is a bounded strongly convex domain with $C^3$-boundary, then the horofunctions h of $(D,k_D)$ are Busemann points and precisely the functions of the form,

$$\begin{aligned} h(z) = \max _{j\in J} h_{\xi _j}(z_j)-\alpha _j, \end{aligned}$$

where $J\subseteq \{1,\ldots ,r\}$ non-empty, $\xi _j\in \partial D_j$ for $j\in J$, and $\min _{j\in J} \alpha _{j}=0$.

Proof

To prove the first assertion let $h\ne h'$ be horofunctions of $(D,k_D)$. As $(D,k_D)$ is a proper geodesic metric space, we know there exist sequences $(w_n)$ and $(z_n)$ in D such that $h_{w_n}\rightarrow h$ and $h_{z_n}\rightarrow h'$. After taking subsequences, we may assume that $w_n\rightarrow \xi \in \partial D$ and $z_n\rightarrow \eta \in \partial D$, since D has a compact norm closure and h and $h'$ are horofunctions.

We claim that $\xi \ne \eta$. To prove this, we need the assumption that $D\subset \mathbb {C}$ is a bounded strongly convex domain with $C^3$-boundary and use results by Abate [1] concerning the so-called small and large horospheres. These are defined as follows: for $R>0$, the small horosphere with centre $\zeta \in \partial D$ (and basepoint $b\in D$) is given by

$$\begin{aligned} \mathcal {E}(\zeta ,R):=\left\{ x\in D:\limsup _{z\rightarrow \zeta } k_D(x,z)-k_D(b,z)<\frac{1}{2}\log R\right\} \end{aligned}$$

and the large horosphere with centre $\zeta \in \partial D$ (and basepoint $b\in D$) is given by

$$\begin{aligned} \mathcal {F}(\zeta ,R):=\left\{ x\in D:\liminf _{z\rightarrow \zeta } k_D(x,z)-k_D(b,z)<\frac{1}{2}\log R\right\} . \end{aligned}$$

We note that the horoballs

$$\begin{aligned} \mathcal {H}(h,\frac{1}{2}\log R)=\left\{ x\in D:\lim _{n\rightarrow \infty } k_D(x,w_n)-k_D(b,w_n)<\frac{1}{2}\log R\right\} \end{aligned}$$

and

$$\begin{aligned} \mathcal {H}(h',\frac{1}{2}\log R)=\left\{ x\in D:\lim _{n\rightarrow \infty }k_D(x,z_n)-k_D(b,z_n)<\frac{1}{2}\log R\right\} \end{aligned}$$

satisfy

$$\begin{aligned} \mathcal {E}(\xi ,R)\subseteq \mathcal {H}(h,\frac{1}{2}\log R)\subseteq \mathcal {F}(\xi ,R)\hbox {and } \mathcal {E}(\eta ,R)\subseteq \mathcal {H}(h',\frac{1}{2}\log R)\subseteq \mathcal {F}(\eta ,R). \end{aligned}$$

It follows from [2, Theorem 2.6.47] (see also [1]) that $\mathcal {E}(\xi ,R)= \mathcal {H}(h,\frac{1}{2}\log R)= \mathcal {F}(\xi ,R)$ and $\mathcal {E}(\eta ,R)= \mathcal {H}(h',\frac{1}{2}\log R)= \mathcal {F}(\eta ,R)$, as D is strongly convex and has $C^3$-boundary. Thus, if $\xi =\eta$, then $h=h'$, since the horoballs, $\mathcal {H}(h,r)$ and $\mathcal {H}(h',r)$ for $r\in \mathbb {R}$, completely determine the horofunctions. This shows that $\xi \ne \eta$. It follows that each horofunction is of the form $h_\xi$, with $\xi \in \partial D$, and hence a Busemann point.

Now suppose that $h_1$ and $h_2$ are horofunctions, with $(z^1_n)$ converging to $h_1$ and $(z^2_n)$ converging to $h_2$. After taking a subsequence, we may assume that $z^1_n\rightarrow \xi _1\in \partial D$ and $z^2_n\rightarrow \xi _2\in \partial D$. We now show that if $\xi _1\ne \xi _2$, then $h_1\ne h_2$. This implies that there is a one-to-one correspondence between the horofunctions of $(D,k_D)$ and $\xi \in \partial D$.

To prove this, we note that as D is strongly convex, D is strictly convex, i.e. for each $\nu \ne \mu$ in $\partial D$, the open straight-line segment $(\nu ,\mu )\subset D$. By [1, Theorem 1.7], we know that $\partial D\cap \mathrm {cl}( \mathcal {F}(h_1,R)) =\{\xi _1\}$ and $\partial D\cap \mathrm {cl}( \mathcal {F}(h_2,R)) =\{\xi _2\}$ for all $R>0$. This implies that $\partial D\cap \mathrm {cl}( \mathcal {H}(h_1,r)) =\{\xi _1\}$ and $\partial D\cap \mathrm {cl}( \mathcal {H}(h_2,r)) =\{\xi _2\}$ for all $r\in \mathbb {R}$. Moreover, from [4, Lemma 5], we know that the straight-line segment $[b,\xi _1]\subset \mathrm {cl}( \mathcal {H}(h_1,0))$. There exists a neighbourhood $W\subset \mathbb {C}^n$ of $\xi _1$ such that $W\cap \mathrm {cl}( \mathcal {H}(h_2,0))=\emptyset$. If we let $w\in [b,\xi _1)\cap W$, then $h_1(w)\le 0$, but $h_2(w)>0$, and hence $h_1\ne h_2$.

Now suppose that $\xi \ne \eta$ in $\partial D$. We know that $\partial D\cap \mathrm {cl}( \mathcal {H}(h_\eta ,0)) =\{\eta \}$ and $\gamma _\xi (t)\not \in \mathrm {cl}( \mathcal {H}(h_\eta ,0))$ for all $t>0$ large. So,

$$\begin{aligned} H(h_\xi ,h_\eta ) = \lim _{t\rightarrow \infty } k_D(\gamma _\xi (t),b) + h_\eta (\gamma _\xi (t))\ge \liminf _{t\rightarrow \infty } k_D(\gamma _\xi (t),b) =\infty , \end{aligned}$$

since $h_\eta (\gamma _\xi (t))\ge 0$ for all t large. This implies that $\delta (h_\xi ,h_\eta ) =\infty$.

The final part follows directly from Theorem 2.3 and Proposition  2.7. $\square$

The proof of Theorem 1.1 is now elementary.

Proof of Theorem 1.1

If $X_j\subset \mathbb {C}^{m_j}$ is a bounded convex domain, then $(X_j,k_{X_j})$ is proper geodesic metric space which contains a geodesic ray by [2, Theorem 2.6.19]. Moreover, if $Y_j\subset \mathbb {C}^{n_j}$ is a bounded strongly convex domain with $C^3$-boundary, then by Lemma 3.3, all the horofunctions of $(Y_j,k_{Y_j})$ are Busemann points and any two distinct Busemann points have infinite detour distance. So, Theorem 1.2 applies and gives the desired result. $\square$

Remark 3.4

I am grateful to Andrew Zimmer for sharing the following observations with me. In the case where $q=1$, Theorem 1.1 can be strengthened and shown in a variety of other ways. Indeed, it was shown by Balogh and Bonk [6] that the Kobayashi distance is Gromov hyperbolic on a strongly pseudo-convex domain with $C^2$-boundary, but the Kobayashi distance on a product domain is clearly not Gromov hyperbolic. This immediately implies Theorem 1.1 for $q=1$ in the more general case where the image domain is strongly pseudo-convex and has $C^2$-boundary.

In fact, if $q=1$ there exists a further strengthening of Theorem 1.1 which only requires the image domain to be strictly convex by using a local argument. The isometric embedding is a locally Lipschitz map with respect to the Euclidean norm, and hence differentiable almost everywhere by Rademacher’s theorem. This implies that the embedding is also an isometric embedding under the Kobayashi infinitesimal metric. On strictly convex domains, the unit balls in the tangent spaces are strictly convex and in product domains, they are not, which yields a contradiction.

Finally, for holomorphic isometric embeddings and $q=1$, Theorem 1.1 can be extended to the case where the image domain is convex with $C^{1,\alpha }$-boundary, see [28, Theorem 2.22].

Looking at the conditions required in Theorem 1.2, it seems likely the regularity conditions on the domains $Y_j$ in Theorem 1.1 can be relaxed considerably. In particular, one may speculate that it suffices to assume that each domain $Y_j$ is strictly convex and has a $C^1$-boundary.