Closed continuations of Riemann surfaces

Let $R_{0}$ be an open Riemann surface of finite genus $g$. If $g=0$, then the general uniformization theorem assures us that $R_{0}$ is conformally equivalent to a domain on the Riemann sphere $\hat{\mathbb{C}}:=\mathbb{C}\cup\{\infty\}$. In 1928 Bochner generalized it to the cases of higher genera in his study [10] on continuations of Riemann surfaces, showing that $R_{0}$ can be realized as a subdomain of a closed Riemann surface of the same genus $g$. While a closed Riemann surface of genus zero is essentially unique, this is not the case for closed Riemann surfaces of positive genus so that there may be two or more closed Riemann surfaces of genus $g$ with a domain conformally equivalent to $R_{0}$ provided that $g>0$. It is then natural to ask into which closed Riemann surfaces of genus $g$ the Riemann surface $R_{0}$ can be conformally embedded.

Heins [22] tackled the problem for $g=1$, and proved in 1953 that the set of closed Riemann surfaces of genus one which are continuations of $R_{0}$ is relatively compact in the moduli space of genus one. Four years later Oikawa [45] formulated the problem in the context of Teichmüller spaces, and discovered that the set of marked closed Riemann surfaces of genus $g$ into which $R_{0}$ can be mapped by a homotopically consistent conformal embedding is compact and connected in the Teichmüller space of genus $g$. In 1987 the second author [56] improved Oikawa’s result in the case of genus one by deducing that Oikawa’s set is in fact a closed disk, which may degenerate to a singleton, with respect to the Teichmüller distance. The authors then have been developing the theory mainly in the framework of Torelli spaces (see, for example, Masumoto [37, 38], Schmieder-Shiba [51, 52, 53] and Shiba [57, 58, 60, 61, 62]). For other results related to conformal embeddings of Riemann surfaces see Bourque [12], Earle-Marden [14], Fehlmann-Gardiner [15], Hamano [19], Hamano-Shiba-Yamaguchi [20], Horiuchi-Shiba [23], Ioffe [25], Ito-Shiba [27, 28], Kahn-Pilgrim-Thurston [30], Masumoto [39, 40], Sasai [49, 50], Shiba [55, 59] and Shiba-Shibata [63, 64]. Applications of conformal embedding theory to hyperbolic geometry and holomorphic mappings are found in Bourque [11] and Masumoto [36, 38, 41, 42, 43].

In the present paper we address the problem for finite open Riemann surfaces of positive genus in the Teichmüller space setting. One of our main purposes is to generalize our previous results in [56] to the cases of higher genera.

To formulate the problem we introduce the category $\mathfrak{F}_{g}$ of marked Riemann surfaces of finite genus $g$ and conformally compatible continuous mappings in §3. Its class of objects consists of equivalence classes ${\bm{S}}=[S,\eta]$, where $S$ is a Riemann surface of genus $g$, not necessarily closed, and $\eta$ is a sense-preserving homeomorphism of $\dot{\Sigma}_{g}$ into $S$, where $\dot{\Sigma}_{g}$ is a fixed surface obtained from a closed oriented topological surface $\Sigma_{g}$ of genus $g$ by deleting one point (see Definition 3.5). A continuous mapping of $\bm{S}$ into another marked Riemann surface ${\bm{S}}^{\prime}=[S^{\prime},\eta^{\prime}]\in\mathfrak{F}_{g}$ is an equivalence class ${\bm{f}}=[f,\eta,\eta^{\prime}]$, where $f$ is a continuous mapping of $S$ into $S^{\prime}$ (see Definition 3.9). The composition ${\bm{f}}^{\prime}\circ{\bm{f}}$ is meaningful only for conformally compatible continuous mappings (see Definition 3.10). Also, a quadratic differential on ${\bm{S}}$ is an equivalence class ${\bm{\varphi}}=[\varphi,\eta]$, where $\varphi$ is a quadratic differential on $S$ (see §6).

The Teichmüller space $\mathfrak{T}_{g}$ of genus $g$ is defined to be the full-subcategory of $\mathfrak{F}_{g}$ whose class of objects is composed of all marked closed Riemann surfaces of genus $g$. Equipped with the Teichmüller distance, $\mathfrak{T}_{g}$ is a metric space homeomorphic to $\mathbb{R}^{2d_{g}}$, where $d_{g}=\max\{g,3g-3\}$. In fact, as is well-known, $\mathfrak{T}_{g}$ is a $d_{g}$-dimensional complex manifold biholomorphic to a bounded domain in $\mathbb{C}^{d_{g}}$. It is quite new to understand the Teichmüller space $\mathfrak{T}_{g}$ as a full-subcategory of $\mathfrak{F}_{g}$. This setting fits our theory because we are required to deal with not only quasiconformal mappings between marked closed Riemann surfaces but also continuous mappings of marked compact bordered Riemann surfaces into marked compact Riemann surfaces.

Let ${\bm{R}}_{0}=[R_{0},\theta_{0}]$ be a marked finite open Riemann surface of positive genus $g$; the fundamental group of $R_{0}$ is finitely generated, or equivalently, $R_{0}$ has finitely many handles and boundary components in the sense of Kerékjártó-Stoïlow (for the definition of a boundary component, see [3, I.36B]). It is said to be analytically finite if $R_{0}$ is conformally equivalent to a Riemann surface obtained from a closed Riemann surface by removing finitely many points. For ${\bm{R}}\in\mathfrak{T}_{g}$ let $\operatorname{CEmb}_{\mathrm{hc}}({\bm{R}}_{0},{\bm{R}})$ be the set of homotopically consistent conformal embeddings of ${\bm{R}}_{0}$ into $\bm{R}$ (see Definition 3.12). By a closed continuation of ${\bm{R}}_{0}$ we mean a pair $({\bm{R}},{\bm{\iota}})$, where ${\bm{R}}\in\mathfrak{T}_{g}$ and ${\bm{\iota}}\in\operatorname{CEmb}_{\mathrm{hc}}({\bm{R}}_{0},{\bm{R}})$. We are concerned with the set $\mathfrak{M}({\bm{R}}_{0})$ of ${\bm{R}}\in\mathfrak{T}_{g}$ for which $\operatorname{CEmb}_{\mathrm{hc}}({\bm{R}}_{0},{\bm{R}})\neq\varnothing$.

The theorem is proved in the final section, where we show a stronger assertion that there is a homeomorphism of $\mathfrak{T}_{g}$ onto $\mathbb{R}^{2d_{g}}$ that maps $\mathfrak{M}({\bm{R}}_{0})$ onto a point or a closed ball. Being a closed Lipschitz domain, $\mathfrak{M}({\bm{R}}_{0})$ satisfies inner and outer cone conditions unless ${\bm{R}}_{0}$ is analytically finite. Actually, it satisfies an outer ball condition. If $g=1$, then its boundary $\partial\mathfrak{M}({\bm{R}}_{0})$ is smooth and a sphere (or rather circle) with respect to the Teichmüller distance. On the other hand, if $g>1$, then the boundary is nonsmooth for some ${\bm{R}}_{0}$ and hence cannot be a sphere with respect to the Teichmüller distance.

For $K\geqq 1$ let $\mathfrak{M}_{K}({\bm{R}}_{0})$ denote the set of ${\bm{R}}\in\mathfrak{T}_{g}$ into which ${\bm{R}}_{0}$ can be mapped by a homotopically consistent $K$-quasiconformal embedding. Thus $\mathfrak{M}_{1}({\bm{R}}_{0})=\mathfrak{M}({\bm{R}}_{0})$. If $K>1$, then $\mathfrak{M}_{K}({\bm{R}}_{0})$ is the closed $(\log K)/2$-neighborhood of $\mathfrak{M}({\bm{R}}_{0})$.

Actually, there is a homeomorphism of $\mathfrak{T}_{g}$ onto $\mathbb{R}^{2d_{g}}$ such that $\mathfrak{M}_{K^{\prime}}({\bm{R}}_{0})$, $K^{\prime}\geqq K$, are mapped onto concentric closed balls. Theorem 1.2 is not a consequence of Theorem 1.1 but a tool for the proof of Theorem 1.1.

For the proofs of Theorems 1.1 and 1.2 we introduce a procedure called self-weldings of marked compact bordered Riemann surfaces ${\bm{S}}=[S,\eta]$, which is also useful to construct various examples of closed continuations. Let $A_{+}({\bm{S}})$ denote the set of nonzero holomorphic quadratic differentials ${\bm{\varphi}}=[\varphi,\eta]$ on $\bm{S}$ with $\varphi\geqq 0$ along the border $\partial S$. We use ${\bm{\varphi}}\in A_{+}({\bm{S}})$ to identify arcs on the border. This operation gives rise to a continuous mapping $\bm{\kappa}$ of $\bm{S}$ onto a marked closed Riemann surface $\bm{R}$ holomorphic and injective on the interior ${\bm{S}}^{\circ}$ of $\bm{S}$ together with a meromorphic quadratic differential $\bm{\psi}$ on $\bm{R}$. The pair $\langle{\bm{R}},{\bm{\kappa}}\rangle$ is referred to as a closed self-welding of $\bm{S}$, and the quadratic differentials $\bm{\varphi}$ and $\bm{\psi}$ are called a welder of $\langle{\bm{R}},{\bm{\kappa}}\rangle$ and the co-welder of $\bm{\varphi}$, respectively. Important are the self-weldings with holomorphic co-welders. Such a self-welding is said to be regular, or $\bm{\varphi}$-regular if we need to refer to the welder $\bm{\varphi}$.

If ${\bm{R}}_{0}$ is nonanalytically finite, that is, it is finite but not analytically finite, then it is considered as the interior of a marked compact bordered Riemann surface $\bm{S}$, possibly, with finitely many points deleted. Each closed self-welding $\langle{\bm{R}},{\bm{\kappa}}\rangle$ of $\bm{S}$ yields a closed continuation $({\bm{R}},{\bm{\kappa}}|_{{\bm{R}}_{0}})$ of ${\bm{R}}_{0}$, called a closed self-welding continuation of ${\bm{R}}_{0}$. A welder of the self-welding is also referred to as a welder of the continuation. If $\langle{\bm{R}},{\bm{\kappa}}\rangle$ is $\bm{\varphi}$-regular, then $({\bm{R}},{\bm{\kappa}}|_{{\bm{R}}_{0}})$ is said to be $\bm{\varphi}$-regular. If this is the case, then ${\bm{\kappa}}|_{{\bm{R}}_{0}}$ is a Teichmüller conformal embedding of ${\bm{R}}_{0}$ into $\bm{R}$, and $\bm{\varphi}$ is an initial quadratic differential of ${\bm{\kappa}}|_{{\bm{R}}_{0}}$.

Let $A_{+}({\bm{R}}_{0})$ be the space of nonzero holomorphic quadratic differentials on ${\bm{R}}_{0}$ that can be extended to elements in $A_{+}({\bm{S}})$. Denote by $A_{L}({\bm{R}})$ the set of ${\bm{\varphi}}=[\varphi,\theta_{0}]\in A_{+}({\bm{R}}_{0})$ such that for any horizontal trajectory $a$ of $\varphi$ on $\partial S$ its $\varphi$-length is at most the half of the $\varphi$-length of the component of $\partial S$ including $a$. Clearly, $A_{L}({\bm{R}}_{0})$ is a proper subset of $A_{+}({\bm{R}}_{0})$.

The theorem shows that the procedure of closed regular self-welding provides us with an explicit method of obtaining all Teichmüller conformal embeddings of ${\bm{R}}_{0}$. Their importance is clarified in the next theorem.

For ${\bm{\varphi}}\in A_{L}({\bm{R}}_{0})$ and ${\bm{R}}\in\mathfrak{M}({\bm{R}}_{0})$ let $\operatorname{CEmb}_{\bm{\varphi}}({\bm{R}}_{0},{\bm{R}})$ be the set of ${\bm{\iota}}\in\operatorname{CEmb}_{\mathrm{hc}}({\bm{R}}_{0},{\bm{R}})$ such that $({\bm{R}},{\bm{\iota}})$ is a closed ${\bm{\varphi}}$-regular self-welding continuation of ${\bm{R}}_{0}$; it can be empty. Set $\operatorname{CEmb}_{\bm{\varphi}}({\bm{R}}_{0})=\bigcup_{{\bm{R}}\in\mathfrak{M}({\bm{R}}_{0})}\operatorname{CEmb}_{\bm{\varphi}}({\bm{R}}_{0},{\bm{R}})$. If $\operatorname{card}\operatorname{CEmb}_{\bm{\varphi}}({\bm{R}}_{0})>1$, then $\bm{\varphi}$ is called exceptional. Let $A_{E}({\bm{R}}_{0})$ be the set of exceptional quadratic differentials in $A_{L}({\bm{R}}_{0})$. Define $\operatorname{CEmb}_{L}({\bm{R}}_{0})=\bigcup_{{\bm{\varphi}}\in A_{L}({\bm{R}}_{0})}\operatorname{CEmb}_{\bm{\varphi}}({\bm{R}}_{0})$ and $\operatorname{CEmb}_{E}({\bm{R}}_{0})=\bigcup_{{\bm{\varphi}}\in A_{E}({\bm{R}}_{0})}\operatorname{CEmb}_{\bm{\varphi}}({\bm{R}}_{0})$.

For ${\bm{\varphi}}\in A_{L}({\bm{R}}_{0})$ let $\mathfrak{M}_{\bm{\varphi}}({\bm{R}}_{0})$ denote the set of ${\bm{R}}\in\mathfrak{M}({\bm{R}}_{0})$ for which $\operatorname{CEmb}_{\bm{\varphi}}({\bm{R}}_{0},{\bm{R}})\neq\varnothing$. If ${\bm{\varphi}}\in A_{L}({\bm{R}}_{0})\setminus A_{E}({\bm{R}}_{0})$, then $\mathfrak{M}_{\varphi}({\bm{R}}_{0})$ is a singleton. Set $\mathfrak{M}_{L}({\bm{R}}_{0})=\bigcup_{{\bm{\varphi}}\in A_{L}({\bm{R}}_{0})}\mathfrak{M}_{\bm{\varphi}}({\bm{R}}_{0})$ and $\mathfrak{M}_{E}({\bm{R}}_{0})=\bigcup_{{\bm{\varphi}}\in A_{E}({\bm{R}}_{0})}\mathfrak{M}_{\bm{\varphi}}({\bm{R}}_{0})$.

We examine the behavior of the extremal length function to prove (i) and (ii) though they also follow from Bourque [11] and Kahn-Pilgrim-Thurston [30]. The set $A_{E}({\bm{R}}_{0})$ is nowhere dense in $A_{L}({\bm{R}}_{0})$ (Theorem 10.3). Nevertheless, quadratic differentials in $A_{E}({\bm{R}}_{0})$ yield abundant boundary points of $\mathfrak{M}({\bm{R}}_{0})$ through closed regular self-weldings, as assertion (iii) claims. If $g=1$, then $A_{E}({\bm{R}}_{0})=\varnothing$. If $g=2$, then there are examples of ${\bm{R}}_{0}$ for which $\mathfrak{M}_{E}({\bm{R}}_{0})$ has a nonempty interior in $\partial\mathfrak{M}({\bm{R}}_{0})$.

Let $\mathscr{MF}(\Sigma_{g})$ be the space of measured foliations on $\Sigma_{g}$. The horizontal foliation of a holomorphic quadratic differential ${\bm{\psi}}=[\psi,\theta]$ on a marked closed Riemann surface ${\bm{R}}=[R,\theta]$ determines an element $\mathcal{H}_{\bm{R}}({\bm{\psi}})$ of $\mathscr{MF}(\Sigma_{g})$ through $\theta$. This defines a homeomorphism $\mathcal{H}_{\bm{R}}$ of the space $A({\bm{R}})$ of holomorphic quadratic differentials on $\bm{R}$ onto $\mathscr{MF}(\Sigma_{g})$ by Hubbard-Masur [24].

Now, for ${\bm{\varphi}}\in A_{L}({\bm{R}}_{0})$, taking a closed $\bm{\varphi}$-regular self-welding continuation $({\bm{R}},{\bm{\iota}})$ of ${\bm{R}}_{0}$ and denoting by $\bm{\psi}$ the co-welder of $\bm{\varphi}$, we set $\mathcal{H}_{{\bm{R}}_{0}}({\bm{\varphi}})=\mathcal{H}_{\bm{R}}({\bm{\psi}})$. Then $\mathcal{H}_{{\bm{R}}_{0}}$ is a well-defined mapping of $A_{L}({\bm{R}}_{0})$ into $\mathcal{MF}(\Sigma_{g})\setminus\{0\}$.

The extremal length function $\operatorname{Ext}$ is a nonnegative continuous function on $\mathfrak{T}_{g}\times\mathscr{MF}(\Sigma_{g})$. Specifically, if $({\bm{R}},\mathcal{F})\in\mathfrak{T}_{g}\times\mathscr{MF}(\Sigma_{g})$, then $\operatorname{Ext}({\bm{R}},\mathcal{F})=\operatorname{Ext}_{\mathcal{F}}({\bm{R}})$ is exactly $\|\mathcal{H}_{\bm{R}}^{-1}(\mathcal{F})\|_{\bm{R}}$, where $\|{\bm{\varphi}}\|_{\bm{S}}$ stands for the $L^{1}$-norm of a quadratic differential $\bm{\varphi}$ on a marked Riemann surface $\bm{S}$.

For $\mathcal{F}\in\mathscr{MF}(\Sigma_{g})\setminus\{0\}$ let $\mathfrak{M}_{\mathcal{F}}({\bm{R}}_{0})$ denote the set of points of $\mathfrak{M}({\bm{R}}_{0})$ at which the restriction of $\operatorname{Ext}_{\mathcal{F}}$ to $\mathfrak{M}({\bm{R}}_{0})$ attains its maximum. It is nonempty since $\mathfrak{M}({\bm{R}}_{0})$ is compact.

Kahn-Pilgrim-Thurston [30, Theorem 1] gives a necessary and sufficient condition for ${\bm{R}}\in\mathfrak{T}_{g}$ to belong to $\mathfrak{M}({\bm{R}}_{0})$ in terms of stretch factors. A stretch factor of a topological embedding of ${\bm{R}}_{0}$ into $\bm{R}$ is defined with the extremal lengths of simple closed multi-curves on ${\bm{R}}_{0}$. Theorem 1.6 (iii) tells us which multi-curves we should take into account to calculate the stretch factor. Specifically, the authors of [30] employ the Jenkins-Strebel quadratic differentials in $A_{+}({\bm{R}}_{0})$ to obtain the stretch factors. Our theorem asserts that those in $A_{L}({\bm{R}}_{0})$ are sufficient.

Besides Theorems 1.3 – 1.6, the Teichmüller geodesic rays induced from the co-welders of elements of $A_{L}({\bm{R}}_{0})$ play an important role in the proof of Theorem 1.2. We then apply analytic properties of the extremal length function to establish Theorem 1.1. Existence of linearly independent elements ${\bm{\varphi}}_{j}\in A_{L}({\bm{R}}_{0})$, $j=1,2$, with $\mathfrak{M}_{{\bm{\varphi}}_{1}}({\bm{R}}_{0})\cap\mathfrak{M}_{{\bm{\varphi}}_{2}}({\bm{R}}_{0})\neq\varnothing$ is one of the biggest obstacles in the proof of Theorem 1.1.

Acknowledgments. This research is supported in part by JSPS KAKENHI Grant Numbers 18K03334 and 22K03556. The authors express sincere thanks to Hideki Miyachi, a brilliant expert on Teichmüller theory, for his invaluable comments and suggestions. Without his help they could not finish the article. They are also grateful to Shuhei Masumoto and Shingo Okuyama for formulating our theory within the scheme of category theory. The two colleagues help the authors write a clearer paper. Last but not least, the authors thank Ken-ichi Sakan for his continuing encouragement.

In the present article we are concerned with conformal embeddings of a Riemann surface into another. We begin with confirming terminology and notation.

Let $X$ and $Y$ be topological spaces. Denote by $\operatorname{Cont}(X,Y)$ the set of continuous mappings of $X$ into $Y$; we avoid employing the usual notation $C(X,Y)$ so as not to confuse it with the set of conformal mappings in the case where $X$ and $Y$ are Riemann surfaces. The subset of homeomorphisms of $X$ onto $Y$ is denoted by $\operatorname{Homeo}(X,Y)$. A topological embedding of $X$ into $Y$ is, by definition, a mapping $f$ of $X$ into $Y$ for which the correspondence $x\mapsto f(x)$ defines an element of $\operatorname{Homeo}(X,f(X))$, where $f(X)$ is endowed with the relative topology induced from the topology of $Y$. We denote by $\operatorname{TEmb}(X,Y)$ the set of topological embeddings of $X$ into $Y$. If $X$ and $Y$ are oriented surfaces, then $\operatorname{Homeo}^{+}(X,Y)$ and $\operatorname{TEmb}^{+}(X,Y)$ denote the sets of sense-preserving elements in $\operatorname{Homeo}(X,Y)$ and $\operatorname{TEmb}(X,Y)$, respectively.

For $f_{k}\in\operatorname{Cont}(X,Y)$, $k=0,1$, we write $f_{0}\simeq f_{1}$ if $f_{0}$ is homotopic to $f_{1}$. Also, for two loops $c_{0}$ and $c_{1}$ on $X$ we use the notation $c_{0}\simeq c_{1}$ to mean that $c_{0}$ is freely homotopic to $c_{1}$. This usage of $\simeq$ is consistent with the previous one as we can regard $c_{0}$ and $c_{1}$ as continuous mappings of the unit circle into $X$.

A surface means a connected $2$-manifold whose topology has a countable base. A surface with boundary is often called a bordered surface or a surface with border. As is well-known, every surface is triangulable, and can be endowed with conformal structure, or complex structure provided that it is orientable.

A Riemann surface is a connected complex manifold of dimension one (see Ahlfors-Sario [3, II.1E] and Strebel [65, Definition 1.1]), and a bordered Riemann surface is a connected 1-dimensional complex manifold with boundary (see [3, II.3A] and [65, Definition 1.2]). By abuse of language we refer to bordered Riemann surfaces also as Riemann surfaces. Thus, when we speak of a Riemann surface, it may be a bordered Riemann surface. A bordered Riemann surface is sometimes called a Riemann surface with border. A Riemann surface without border means a Riemann surface which is not a bordered Riemann surface. A Riemann surface without border is called closed (resp. open) if it is compact (resp. noncompact). Thus a compact Riemann surface is either a closed Riemann surface or a compact bordered Riemann surface. By a torus we mean a closed Riemann surface of genus one.

A holomorphic mapping of a Riemann surface into another is called a conformal embedding if it is a topological embedding at the same time, where holomorphic mappings of a bordered Riemann surface are supposed to be analytic on the border. A biholomorphic mapping of a Riemann surface onto another is also referred to as a conformal homeomorphism. Conformal homeomorphisms are conformal embeddings. Two Riemann surfaces are said to be conformally equivalent to each other if there is a conformal homeomorphism of one onto the other.

For Riemann surfaces $S$ and $R$ the sets of conformal embeddings of $S$ into $R$ and conformal homeomorphisms of $S$ onto $R$ are denoted by $\operatorname{CEmb}(S,R)$ and $\operatorname{CHomeo}(S,R)$, respectively. By a conformal automorphism of $S$ we mean a conformal homeomorphism of $S$ onto itself. Let $\operatorname{Aut}(S)$ be the group of conformal automorphisms of $S$. Thus $\operatorname{Aut}(S)=\operatorname{CHomeo}(S,S)$. Its subgroup consisting of $\kappa\in\operatorname{Aut}(S)$ satisfying $\kappa\simeq\operatorname{id}_{S}$ is denoted by $\operatorname{Aut}_{0}(S)$.

Two continuations $(R_{j},\iota_{j})$, $j=1,2$, of $S$ are defined to be equivalent to each other if there is $\kappa\in\operatorname{CHomeo}(R_{1},R_{2})$ such that $\iota_{2}=\kappa\circ\iota_{1}$. A continuation $(R,\iota)$ of $S$ is called closed (resp. open, compact) if $R$ is closed (resp. open, compact). If $\iota(S)$ is dense in $R$, then we say that $(R,\iota)$ is a dense continuation of $S$. A continuation $(R,\iota)$ of $S$ is said to be genus-preserving if each connected component of $R\setminus\iota(S)$ is topologically embedded into $\mathbb{C}$. In the case where $S$ is of finite genus, a continuation $(R,\iota)$ of $S$ is genus-preserving if and only if $R$ is of the same genus as $S$. Observe that if $S$ is a bordered Riemann surface, then there is a genus-preserving open continuation $(R,\iota)$ of $S$ such that $\iota(S)$ is a retract of $R$. If $(R,\iota)$ is a continuation of $S$ and $(R^{\prime},\iota^{\prime})$ is a continuation of $R$, then $(R^{\prime},\iota^{\prime}\circ\iota)$ is a continuation of $S$.

Let $f$ be a quasiconformal embedding of a Riemann surface $S$ without border into a Riemann surface. If $z$ is a local parameter around a point $p\in S$ and $w$ is a local parameter around $f(p)$, then with respect to these parameters $f$ is regarded as a complex function $w=f(z)$. The quantity $\bar{\partial}f/\partial f=(f_{\bar{z}}\,d\bar{z})/(f_{z}\,dz)$ does not depend on a particular choice of $w$ and defines a measurable $(-1,1)$-form $\mu_{f}$ on $S$ called the Beltrami differential of $f$. Then $|\mu_{f}|$ is a measurable function on $S$, whose $L^{\infty}$-norm $\|\mu_{f}\|_{\infty}$ is less than one. The maximal dilatation $K(f)$ of $f$ is defined by $K(f)=(1+\|\mu_{f}\|_{\infty})/(1-\|\mu_{f}\|_{\infty})$. Denote by $\operatorname{QCEmb}(S,R)$ and $\operatorname{QCHomeo}(S,R)$ the sets of quasiconformal embeddings of $S$ into $R$ and quasiconformal homeomorphisms of $S$ onto $R$, respectively.

Let $h_{j}\in\operatorname{QCHomeo}(S,R_{j})$, $j=1,2$, where $S$ and $R_{j}$ are Riemann surfaces without border. If $\mu_{h_{1}}=\mu_{h_{2}}$ almost everywhere on $S$, then $h_{2}\circ h_{1}^{-1}\in\operatorname{CHomeo}(R_{1},R_{2})$. For any measurable $(-1,1)$-form $\mu$ on $S$ with $\|\mu\|_{\infty}<1$ there is a quasiconformal homeomorphism $h$ of $S$ onto a Riemann surface without border for which $\mu_{h}=\mu$ almost everywhere on $S$.

It is convenient to define quasiconformal embeddings also on bordered Riemann surfaces. Let $S$ be a bordered Riemann surface. A topological embedding $f$ of $S$ into another Riemann surface $R$ is called quasiconformal if there are continuations $(S^{\prime},\iota^{\prime})$ and $(R^{\prime},\kappa^{\prime})$ of $S$ and $R$, respectively, where $S^{\prime}$ and $R^{\prime}$ are Riemann surfaces without border, together with $f^{\prime}\in\operatorname{QCEmb}(S^{\prime},R^{\prime})$ such that $\kappa^{\prime}\circ f=f^{\prime}\circ\iota^{\prime}$. We can then speak of the maximal dilatation $K(f)$. Note that $K(f^{\prime})\geqq K(f)$. Each component of the border of $S$ is mapped by $f$ onto a simple arc or loop of vanishing area.

Let $K$ be a real constant with $K\geqq 1$. A quasiconformal embedding is called $K$-quasiconformal if its maximal dilatation does not exceed $K$. Conformal embeddings are $1$-quasiconformal, and vice versa.

A subset $S^{\prime}$ of a Riemann surface $S$ is called a subsurface of $S$ if it carries its own conformal structure such that the inclusion mapping of $S^{\prime}$ into $S$ is a conformal embedding of $S^{\prime}$ into $S$. Each component of the border of $S^{\prime}$, if any, is an analytic curve on $S$. Subdomains of $S$, or nonempty connected open subsets of $S$, are subsurfaces of $S$.

Let $S$ be a bordered Riemann surface. We denote its border by $\partial S$, and set $S^{\circ}=S\setminus\partial S$, called the interior of $S$. Note that $S^{\circ}$ is a subsurface of $S$ and that $\partial S$, which is also a topological boundary of $S^{\circ}$, is a 1-dimensional real analytic submanifold of $S$. If $\iota$ denotes the inclusion mapping of $S^{\circ}$ into $S$, then we call $(S,\iota)$ the canonical continuation of $S^{\circ}$. In the case where $S$ is a Riemann surface without border, we define $\partial S=\varnothing$ and $S^{\circ}=S$ for convenience, and call $S^{\circ}$ the interior of $S$.

A Riemann surface $S$ is said to be analytically finite if there is a closed continuation $(R,\iota)$ of $S$ such that $R\setminus\iota(S)$ is a finite set, that is, $\operatorname{card}(R\setminus\iota(S))<\aleph_{0}$, where $\operatorname{card}E$ denotes the cardinal number of a set $E$. A Riemann surface is said to be a nonanalytically finite if it is finite but not analytically finite.

Analytically finite Riemann surfaces are finite Riemann surfaces without border. Closed Riemann surfaces are analytically finite. If $S$ is an analytically finite Riemann surface and $h$ is a quasiconformal homeomorphism of $S$ onto another Riemann surface $R$, then $R$ is also analytically finite.

Clearly, $(\breve{S},\breve{\iota})$ is a dense continuation of $S$. To obtain a natural compact continuation of $S$ we have only to consider the case where $S$ is nonanalytically finite. Then the universal covering Riemann surface of $S$ is conformally equivalent to the upper half plane $\mathbb{H}:=\{z\in\mathbb{C}\mid\operatorname{Im}z>0\}$. The covering transformation group $\Gamma$ is a torsion-free Fuchsian group of the second kind keeping $\mathbb{H}$ invariant, and $\mathbb{H}/\Gamma$ is conformally equivalent to $S$. Let $\Omega(\Gamma)$ denote the region of discontinuity of $\Gamma$, and set $S^{\prime}=(\bar{\mathbb{H}}\cap\Omega(\Gamma))/\Gamma$, where $\bar{\mathbb{H}}$ is the closure of $\mathbb{H}$ in the extended complex plane $\hat{\mathbb{C}}:=\mathbb{C}\cup\{\infty\}$. If $S^{\prime}$ is compact, then we set $\breve{S}=S^{\prime}$. Otherwise, $S^{\prime}$ has finitely many punctures and we let $\breve{S}$ be the Riemann surface $S^{\prime}$ with the punctures filled in. In any case we have a conformal embedding $\breve{\iota}$ of $S$ into $\breve{S}$ to obtain a desired continuation $(\breve{S},\breve{\iota})$. As is easily verified, two natural compact continuations of $S$ are equivalent to each other.

Let $R_{0}$ be a Riemann surface of finite genus. Then it is conformally embedded into a closed Riemann surface of the same genus by Bochner [10, Satz V], which gives rise to a genus-preserving closed continuation of $R_{0}$. We are interested in the set of closed Riemann surfaces of the same genus as $R_{0}$ into which $R_{0}$ can be conformally embedded, and will work in the context of Teichmüller theory.

There are several equivalent definitions of Teichmüller spaces for closed Riemann surfaces. In the present article we adopt the following definition. The point is how we should define continuous mappings between marked Riemann surfaces.

We start with introducing the category $\mathfrak{F}_{g}$ of marked Riemann surfaces of genus $g$ and conformally compatible continuous mappings. Fix a closed oriented surface $\Sigma_{g}$ of positive genus $g$, and remove one point from $\Sigma_{g}$ to obtain a once-punctured closed oriented surface $\dot{\Sigma}_{g}$. For the sake of definiteness, using the notations in Example 2.3, we set $\Sigma_{1}=T_{i}=T_{\sqrt{-1}}$ and $\dot{\Sigma}_{1}=\Sigma_{1}\setminus\{\Pi_{i}((1+i)/2)\}$.

If $\eta$ is a $g$-handle mark of $S$, then $\eta(\dot{\Sigma}_{g})\subset S^{\circ}$. Thus $\eta$ is also considered as a $g$-handle mark of $S^{\circ}$.

A marked Riemann surface ${\bm{S}}=[S,\eta]$ is called a marked bordered Riemann surface or a marked Riemann surface with border if $S$ is a bordered Riemann surface. Otherwise, $\bm{S}$ is referred to as a marked Riemann surface without border. If $S$ is closed (resp. open, compact), then $\bm{S}$ is called closed (resp. open, compact). We also say that $\bm{S}$ is a marked closed (resp. open, compact) Riemann surface. The meaning of a marked torus is obvious. If $S$ is finite (resp. analytically finite, nonanalytically finite), then $\bm{S}$ is said to be finite (resp. analytically finite, nonanalytically finite). Since $\eta(\dot{\Sigma}_{g})\subset S^{\circ}$, regarding $\eta$ as a topological embedding of $\dot{\Sigma}_{g}$ into $S^{\circ}$, we obtain a marked Riemann surface ${\bm{S}}^{\circ}=[S^{\circ},\eta]$ without border, which will be called the interior of $\bm{S}$.

We show two propositions to verify that the marked closed Riemann surfaces defined above are essentially the same as the known ones. The second proposition also plays a fundamental role in our investigations on closed continuations.