On Geodesic Segments in the Infinitesimal Asymptotic Teichmüller Spaces

Abstract

Let be the infinitesimal asymptotic Teichmüller space of a Riemann surface of infinite type. It is known that is the quotient Banach space of the infinitesimal Teichmüller space , where is the dual space of integrable quadratic differentials. The purpose of this paper is to study the nonuniqueness of geodesic segment joining two points in . We prove that there exist infinitely many geodesic segments between the basepoint and every nonsubstantial point in the universal infinitesimal asymptotic Teichmüller space by constructing a special degenerating sequence.

1. Introduction

Let be a hyperbolic Riemann surface, that is, a Riemann surface with universal covering surface which is conformally equivalent to the open unit disk . Denote by the Banach space of Beltrami differentials on with finite -norms.

Let be the space of integrable holomorphic quadratic differentials on with -norms: Denote by the unit sphere of and by the set of all degenerating sequences of . A sequence of quadratic differentials in is said to be a degenerating sequence if and locally uniformly on as .

Two elements are called infinitesimally Teichmüller equivalent, if We denote by the infinitesimal Teichmüller equivalence class of . The infinitesimal Teichmüller space is the set of all infinitesimal Teichmüller equivalence classes of ’s in . The point is called the basepoint of .

A Beltrami differential is said to be vanishing at infinity, if, for each , there is a compact set in such that . Denote by the set of all such vanishing ’s and by the set of all infinitesimal Teichmüller equivalence classes of ’s in .

Two elements are called infinitesimally asymptotically equivalent, if there exists such that We denote by the infinitesimal asymptotic equivalence class of . The infinitesimal asymptotic Teichmüller space is the set of all infinitesimal asymptotic equivalence classes of ’s in . The point is called the basepoint of .

It is known that and are the tangent spaces to the classical Teichmüller space and asymptotic Teichmüller space at their basepoint, respectively (see [1]). For further results and properties about Teichmüller theory, we refer to the papers [1–5] and the books [6–8].

The notion of geodesics plays an important role in the study of the geometry of Teichmüller theory. A geodesic in a metric space is a continuous curve such that for any subarc its length is equal to the distance between its two endpoints. The existence and uniqueness of geodesic between two points in various spaces have been discussed for a long time (see [6, 9–15]). Given two points in infinitesimal Teichmüller space , it is shown in [6] that there exists at least one geodesic joining them. Furthermore, it is proved in [14] that there exists precisely one geodesic segment joining the basepoint and in if and only if contains a uniquely extremal Beltrami coefficient with constant modulus.

The existence of geodesic joining two given points in has been proved in [1, 6]; however, the uniqueness of geodesic is unknown. In this paper, we will prove the nonuniqueness of geodesics in the universal infinitesimal asymptotic Teichmüller space by constructing a special degenerating sequence in .

The structure of this paper is as follows. Section 2 is devoted to setting up the notations and some results we need. In Section 3, a special sequence degenerating towards a boundary point of the unit circle is constructed. In Section 4, it is proved that there are infinitely many geodesics joining with the basepoint in when is of a nonsubstantial point using the important lemma (Lemma 2) and the constructed sequence in the previous section.

2. Preliminaries

In this section, we recall some notions and basic results from the Teichmüller theory. For more details, we refer to the book [6] and the paper [1].

By the Hahn-Banach and Riesz representation theorem, every element in the dual space of the Banach space of all integrable holomorphic quadratic differentials can be represented as where is a Beltrami differential in . So there is natural one-to-one correspondence between the infinitesimal Teichmüller space and the dual space . Thus, in what follows, the infinitesimal Teichmüller equivalence classes of Beltrami differentials on and complex linear functionals on are used for points of alternately.

For every , the infinitesimal extremal dilatation and the infinitesimal boundary dilatation of are defined as respectively, where is called infinitesimal extremal if . It is shown in [6] that is a Banach space with the infinitesimal Teichmüller norm

The infinitesimal Teichmüller distance between two points and in is defined in [14] where and represent and , respectively.

Let ; be the quotient mapping from the tangent space to the tangent space . and in represent the same point in if . Thus, the infinitesimally asymptotic equivalence classes of Beltrami differentials are in one-to-one correspondence with the elements of and represents .

For any , we define the quotient norm on the quotient space as It is known in [6] that . is called infinitesimally asymptotically extremal if . Furthermore, is a Banach space with the standard seminorm (see [6]) where represents . In particular, if , it holds

The infinitesimal asymptotic Teichmüller distance between two points and in is defined as where and represent and , respectively.

For any , is an infinitesimal extremal representative of if and only if it has a so-called Hamilton sequence, namely, a sequence , such that Similarly, for any , is an infinitesimal asymptotically extremal representative of [1, 6] if and only if there exists an asymptotic Hamilton sequence of , namely, a degenerating sequence , such that

Let be the unit disk in the extended complex plane and let be the unit circle. In the following part, we consider some results about the infinitesimal local boundary dilatation of in the tangent space to the universal Teichmüller space.

Set and . For any , the infinitesimal local boundary dilatation of at is defined as where

For any , we define and then . It is proved by Lakic [16] that A point with is said to be an infinitesimal substantial boundary point for . (or ) is called an infinitesimal substantial point in , if every is an infinitesimal substantial boundary point for (or ); otherwise, (or ) is called an infinitesimal nonsubstantial point.

The following lemma can be obtained in [17] by Fehlmann and Sakan.

Lemma 1. For any , let be an infinitesimal extremal representative of . Suppose there is a point which is not a substantial boundary point of . Then, there is an open interval , , and a domain such that, for any degenerating Hamilton sequence of , one has

3. Constructing a Special Sequence Degenerating towards a Boundary Point

In this section, we will construct a special sequence degenerating towards a boundary point . The method used here is similar to that in [18] while the sequence degenerates towards the whole boundary in this paper.

Let and . A degenerating sequence is said to degenerate towards if, for any neighbourhood of with , Then, for any and , there exists a positive integer , such that holds for every . Since , , there exists a positive number satisfying By the definition of degenerating sequence and (26), there exists such that

Choose a positive number such that where . It follows from (29) and (30) that

By induction, we obtain a subsequence of and a positive number sequence with and such that where .

Without loss of generality, from now on, we write instead of for simplicity. Let From (33), it is easy to see that this series is uniformly convergent in every compact subset of . So is a holomorphic quadratic differential on .

Noting that and by (34), we get that is, Moreover, by the second formula of (37), we have

By simple calculation, it follows from (35), (37), and (39) that Then, Furthermore, from (34) and (41), we have

4. Nonuniqueness of Geodesics Joining Every Infinitesimal Nonsubstantial Point with the Basepoint in

For every , it is known in [1] that there exists a representative such that . Thenmeans is an infinitesimal extremal representative of , and implies is an infinitesimal asymptotic extremal representative of . Set and It is clear that and ; moreover, is a geodesic segment joining and in .

Let and . In this section, we will discuss the nonuniqueness of geodesic segments between and in when is an infinitesimal nonsubstantial point; that is, there exists a point with .

We need the following important lemma.

Lemma 2. Let and . Then, for any given , there exists an infinitesimal asymptotic extremal representative of such that

Proof. Suppose is an infinitesimal substantial boundary point for ; that is, . It is known in [6] that there exists an infinitesimal asymptotic extremal representative such that . We conclude that (46) holds since Otherwise, suppose . For any , without loss of generality, we assume that . By the definition of the infinitesimal local boundary dilatation, there exists a Beltrami differential representing such that Moreover, by the definition of , there exists such that Let be an infinitesimal extremal and asymptotic extremal representative of ; that is, , and let be an asymptotic Hamilton sequence of . By Lemma 1 ( sufficiently small), we have So which means There exists a boundary point outside satisfying so we have . Let be the characteristic function and . Then Since we get due to (52). It follows from (18) and (54) that So is an infinitesimal asymptotically extremal Beltrami differential in its equivalence class and .
Define It is not hard to verify that is an infinitesimal asymptotically extremal representative of , and by (48) and (49), . This completes the proof of Lemma 2.

Theorem 3. For every , if is a nonsubstantial point, that is, for some point , then there exist infinitely many geodesic segments connecting and the basepoint in .

Proof. Let . By Lemma 2, there exists an infinitesimal asymptotic extremal representative of such that So there exists a positive number and a neighbourhood of such that Set , and . Since is infinitesimally asymptotically extremal, . From (18), there exists an asymptotic Hamilton sequence such that Furthermore, by Lemma 1, we have for sufficiently small .
Let be the characteristic function of and let be the special sequence degenerating towards constructed as above. For every and , let where is a holomorphic quadratic differential on . Clearly, and .
Now we show that is a geodesic segment in . Let . We discuss the infinitesimally asymptotic equivalence class By Lemma 1, it is easy to calculate that Since we obtain that is infinitesimally asymptotically extremal in its equivalence class. From (16), we have which implies that is a geodesic segment joining and in .
Now we prove that, for , and are two different geodesics joining and in . Otherwise, suppose , when . Then From (15), for the special sequence degenerating towards constructed as above, it yields On the other hand, there exists a positive number sequence corresponding to with and such that It follows from (38) that So Furthermore, It follows from (41) and (42) that Therefore, By (71)–(76), we conclude that This is a contradiction with (70), which implies for if .
Thus, we have constructed infinitely many geodesics () joining and in .

The situation on the geodesics joining an infinitesimal substantial point with the basepoint is not clear. We conjecture that there exist infinitely many geodesics between an infinitesimal substantial point and the basepoint in .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The research is partially supported by the National Natural Science Foundation of China (Grants nos. 11371045, 11301248, and 11271175).