On critical extremal length for the existence of holomorphic mappings of once-holed tori
© Masumoto; licensee Springer 2013
Received: 15 December 2012
Accepted: 22 May 2013
Published: 5 June 2013
Let be the set of marked once-holed tori which allows a holomorphic mapping into a given Riemann surface with marked handle. We compare it with the subset of marked once-holed tori X such that there is a holomorphic mapping for which the cardinal numbers of , , are bounded. We show that while is a proper subset of apart from a few exceptions, their critical extremal lengths are identical.
Dedicated to Professor Hari M Srivastava
Let R be a Riemann surface of positive genus. By a mark of handle of R we mean an ordered pair of simple loops a and b on R whose geometric intersection number is equal to one. The pair is called a Riemann surface with marked handle.
Let be another Riemann surface with marked handle, where . If is holomorphic and maps a and b onto loops freely homotopic to and on , respectively, then we say that f is a holomorphic mapping of Y into and use the notation . If, in addition, is conformal, that is, if is holomorphic and injective, then is called conformal.
A noncompact Riemann surface of genus one with exactly one boundary component is called a once-holed torus. A marked once-holed torus means a once-holed torus with marked handle. Let denote the set of marked once-holed tori, where two marked once-holed tori are identified with each other if there is a conformal mapping of one onto the other. It is a three-dimensional real analytic manifold with boundary (see [, §7]).
For later use, we introduce some notations. Let ℍ stand for the upper half-plane: . For , let denote the additive group generated by 1 and τ. Then is a torus, that is, a compact Riemann surface of genus one. The two oriented segments and are projected onto simple loops and forming a mark of handle of . Set . For , we define , where is the natural projection. Then is a once-holed torus. We choose a mark of handle of so that the inclusion mapping of into is a conformal mapping of into . The correspondence defines a bijection of onto (see, for example, ). In other words, every marked once-holed torus is realized as a horizontal slit domain of a torus with marked handle, or a marked torus, uniquely up to conformal automorphisms of the marked torus.
if , then for any l, while
if , then for some l,
where . If is a marked torus, then and hence . Otherwise, by [, Theorem 1 and Proposition 1].
if , then for any l, while
if , then for some l.
We first establish the following theorem.
Theorem 1 If is not a marked torus or a marked once-holed torus, then is a proper subset of .
Nevertheless, the sign of equality actually holds in (1).
Theorem 2 For any marked Riemann surface , the equality holds.
The proofs of Theorems 1 and 2 will be given in the next section.
We begin with the proof of Theorem 1. Let , where , be a Riemann surface with marked handle which is not a marked torus or a marked once-holed torus. We consider the loops and as elements of the fundamental group of . Let be the covering Riemann surface of corresponding to the subgroup of generated by and . Since is not a torus, is a once-holed torus. We choose a mark of handle of so that the natural projection is a holomorphic mapping of the marked once-holed torus onto . Then is an element of .
Let f be an arbitrary holomorphic mapping of into . Since it maps and onto loops freely homotopic to and , respectively, it is lifted to a holomorphic mapping of into itself satisfying . By Huber [, Satz II] (see also Marden-Richards-Rodin [, Theorem 5]), we infer that is a conformal automorphism of . Since is not a torus or a once-holed torus, we conclude that and hence . This completes the proof of Theorem 1.
For the proof of Theorem 2, we make a remark. Let , where , be a marked once-holed torus. Then the extremal length of the free homotopy class of a is called the basic extremal length of X. Note that (see [, Proposition 1]).
Now, take an arbitrary with . Then, for some , there is a holomorphic mapping f of into . Recall that is the horizontal slit domain of the torus . Choose a canonical exhaustion of so that each is a once-holed torus including the loops in . Since the inclusion mapping is a conformal mapping of the marked once-holed torus into , the restriction of f to is a holomorphic mapping of into . As is relatively compact in , we know that . Consequently, belongs to .
Theorem 2 has been proved.
3 Topological relations between and
The arguments in the proof of Theorem 2 easily lead us to the following theorem.
Theorem 3 The closure of is identical with .
Proof We begin with recalling a global coordinate system on the space of marked once-holed tori. Let be a marked once-holed torus, where . Observe that is a mark of handle of T. Also, if c is a simple loop homotopic to , then is another mark of handle of T. Set and . Then the basic extremal lengths of X, and define a global coordinate system on . In fact, we introduce a real analytic structure into so that the mapping is a real analytic diffeomorphism of into (see ).
Now, let be an arbitrary element of . Take a canonical exhaustion of T for which each is a once-holed torus including the loops in χ, and set . Since for some and , the proof of Theorem 2 shows that the basic extremal length tends to as . By changing marks of handles, we infer that and converge to and , respectively, and hence that as . Since each belongs to , the marked once-holed torus X belongs to the closure of . We thus obtain . Because is closed and includes , we conclude that . □
Since and are (noncompact) domains with Lipschitz boundary by , we see that Theorem 3 is an improvement of Theorem 2. Also, we obtain the following corollary.
Corollary 1 The interiors of and coincide with each other.
If is a marked torus, then is identical with (see ). Hence so is by Corollary 1.
This research is supported in part by JSPS KAKENHI Grant Number 22540196. The author is grateful to the referees for their invaluable comments.
- Masumoto M: Conformal mappings of a once-holed torus. J. Anal. Math. 1995, 66: 117–136. 10.1007/BF02788820MathSciNetView ArticleMATHGoogle Scholar
- Shiba M: The moduli of compact continuations of an open Riemann surface of genus one. Trans. Am. Math. Soc. 1987, 301: 299–311. 10.1090/S0002-9947-1987-0879575-2MathSciNetView ArticleMATHGoogle Scholar
- Masumoto, M: Holomorphic mappings of once-holed tori. PreprintGoogle Scholar
- Masumoto M: Conformal and holomorphic mappings of once-holed tori. Global J. Math. Sci. 2012, 1: 24–30.Google Scholar
- Masumoto M: Holomorphic mappings and basic extremal lengths of once-holed tori. Nonlinear Anal. 2009, 71: e1178-e1181. 10.1016/j.na.2009.01.136MathSciNetView ArticleMATHGoogle Scholar
- Huber H: Über Analytische Abbildungen Riemannscher Flächen in sich. Comment. Math. Helv. 1953, 27: 1–73. 10.1007/BF02564552MathSciNetView ArticleMATHGoogle Scholar
- Marden A, Richards I, Rodin B: Analytic self-mappings of Riemann surfaces. J. Anal. Math. 1967, 18: 197–225. 10.1007/BF02798045MathSciNetView ArticleMATHGoogle Scholar
- Masumoto M: Estimates of the Euclidean span for an open Riemann surface of genus one. Hiroshima Math. J. 1992, 14: 573–582.MathSciNetMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.