It is proved that if f is a Teichmuller self-mapping of the unit disk with a holomorphic quadratic deferential and satisfies the growth condition m(ψ，r)= o((1 -r)-), r→1, for any s>1, then f is extremal, and the...It is proved that if f is a Teichmuller self-mapping of the unit disk with a holomorphic quadratic deferential and satisfies the growth condition m(ψ，r)= o((1 -r)-), r→1, for any s>1, then f is extremal, and there exists a sequence {tn}, 0<tn<1, /lim, tn =1, such that {(tnz)} is a Hamilton sequence. It is the precision of a theorem of Reich-Strebel in 1974, and gives a fairly satisfactory answer to a question of Reich in 1988.展开更多
Let S be a Riemann surface with genus p and n punctures. Assume that 3p- 3 ＋ n 〉 0 and n 〉 1. Let a be a puncture of S and let S = S∪{α}. Then all mapping classes in the mapping class group Mods that fixes the pu...Let S be a Riemann surface with genus p and n punctures. Assume that 3p- 3 ＋ n 〉 0 and n 〉 1. Let a be a puncture of S and let S = S∪{α}. Then all mapping classes in the mapping class group Mods that fixes the puncture a can be projected to mapping classes of Mod$ under the forgetful map. In this paper the author studies the mapping classes in Mods that can be projected to a given hyperbolic mapping class in Mode.展开更多
The purpose of this paper is to give a relatively elementary and direct proof of the Delta Inequality, which plays a very important role in the study of the extremal problem of quasiconformal mappings.
In 1981, E. Reich established a theorem which stated a sufficient condition for a quasiconformal mapping to be a unique extremal mapping with the given boundary value, and asked whether or not such a uniquely extremal...In 1981, E. Reich established a theorem which stated a sufficient condition for a quasiconformal mapping to be a unique extremal mapping with the given boundary value, and asked whether or not such a uniquely extremal mapping was a Teichmüller mapping in his remark. In this paper, the above open problem was solved and consequently the theorem was strengthened.展开更多
The boundary value problem for harmonic maps of the Poincare disc is discussed. The emphasis is on the non-smoothness of the given boundary values in the problem. Let T . be a subspace of the universal Teichmülle...The boundary value problem for harmonic maps of the Poincare disc is discussed. The emphasis is on the non-smoothness of the given boundary values in the problem. Let T . be a subspace of the universal Teichmüller space, defined as a set of normalized quasisymmetric homeomorphisms h of the unit circle S onto itself where h admits a quasiconformal extension to the unit disc D with a complex dilatation μ satisfyingwhere ρ(z)|dz|2 is the Poincare metric of D. Let B . be a Banach space consisting of holomorphic quadratic differentials φ in D with normsIt is shown that for any given quasisymmetric homeomorphism h : S1→S1∈ T . , there is a unique quasiconformal harmonic map of D with respect to the Poincare metric whose boundary corresponding is h and the Hopf differential of such a harmonic map belongs to B .展开更多
It is well known that certain isotopy classes of oseudo-Anosov maos on a Riemann surface S of non-excluded type can be defined through Dehn twists tα and tβ along simple closed geodesics α and β on S,respectively....It is well known that certain isotopy classes of oseudo-Anosov maos on a Riemann surface S of non-excluded type can be defined through Dehn twists tα and tβ along simple closed geodesics α and β on S,respectively. Let G be the corresponding Fuchsian group acting on the hyperbolic plane H so that H/G≌S.For any point α∈S,define S = S／{α}.In this article, the author gives explicit parabolic elements of G from which he constructs pseudo-Anosov classes on S that can be projected to a given pseudo-Anosov class on S obtained from Thurston＇s construction.展开更多
LET D be the unit disk in the complex plane C and f be a sense preserving quasisymmetrichomeomorphism of D onto itself. Denote by Q a quadrilateral D （z<sub>1</sub>, z<sub>2</sub>, z<sub>...LET D be the unit disk in the complex plane C and f be a sense preserving quasisymmetrichomeomorphism of D onto itself. Denote by Q a quadrilateral D （z<sub>1</sub>, z<sub>2</sub>, z<sub>3</sub>, z<sub>4</sub> ） with do-main D and vertices z<sub>1</sub>, z<sub>2</sub>, z<sub>3</sub>, z<sub>4</sub> ∈D, and by M（Q） its conformal modulus. We are in-展开更多
A constantK 0 (m) (h) is introduced for every quasisymmetric mappingh of the unit circle and every integerm≥4 which contains the constantK 0(h) (indicated by the change in module of the quadrilaterals with vertices o...A constantK 0 (m) (h) is introduced for every quasisymmetric mappingh of the unit circle and every integerm≥4 which contains the constantK 0(h) (indicated by the change in module of the quadrilaterals with vertices on the circle) as a special case. A necessary and sufficient condition is established forK 0 (m) (h) =K 1(h). It is shown that there are infinitely many quasisymmetric mappings of the unit circle having the property thatK 0 (m) (h)<K 1(h), wherek 1(h) is the maximal dilatation ofh.展开更多
Two necessary and sufficient conditions for the validity of the conjecture K 0(h)=K 1(h) are given, which are independent of the complex dilatations of extremal quasiconformal mappings, where K 0(h) is the maximal con...Two necessary and sufficient conditions for the validity of the conjecture K 0(h)=K 1(h) are given, which are independent of the complex dilatations of extremal quasiconformal mappings, where K 0(h) is the maximal conformal modulus dilatation of the boundary homeomorphism h, K 1(h) is the maximal dilatation of extremal quasiconformal mappings that agree with h on the boundary. In addition, when the complex dilatation of an extremal quasiconformal mapping is known, the proof of the result simplifies Reich and Chen Jixiu-Chen Zhiguo’s result.展开更多
This paper studies the subset of the non-Strebel points in the universal Teichmüller spaceT. Let z0 ∈ Δ be a fixed point. Then we prove that for every non-Strebel pointh, there is a holomorphic curve γ: [0, 1]...This paper studies the subset of the non-Strebel points in the universal Teichmüller spaceT. Let z0 ∈ Δ be a fixed point. Then we prove that for every non-Strebel pointh, there is a holomorphic curve γ: [0, 1] →T withh as its initial point satisfying the following conditions. (1) The curve γ is on a sphere centered at the base-point ofT, i.e.d T (id, γ(t))=d T (id, h), (t∈[0, 1]). (2) For everyt ∈ (0,1], the variability set Vγ(t)[z0] of γ(t) has non-empty interior, i.e. .展开更多
Let QS* (S 1) be the space of quasisymmetric homeomorphisms of the unit circle such that the corresponding subspace of the universal Teichmu¨ller space has Weil-Petersson metric.In this paper we give a necessary ...Let QS* (S 1) be the space of quasisymmetric homeomorphisms of the unit circle such that the corresponding subspace of the universal Teichmu¨ller space has Weil-Petersson metric.In this paper we give a necessary condition for a quasisymmetric homeomorphism to belong to QS *(S 1) from the aspect of cross-ratio distortion.展开更多
LetT be the universal Teichmüller space viewed as the set of all normalized quasisymmetric homeomorphism of the unit circleS 1=?Δ. Denote byV h [z 0] the variability set ofz 0 with respect toh∈T. The following ...LetT be the universal Teichmüller space viewed as the set of all normalized quasisymmetric homeomorphism of the unit circleS 1=?Δ. Denote byV h [z 0] the variability set ofz 0 with respect toh∈T. The following is proved: Leth 0 be a point ofT. Suppose thatμ 0 is an arbitrarily given extremal Beltrami differential ofh 0 andf 0: μ→μ is a quasiconformal mapping with the Beltrami coefficientμ 0 andf 01s=h 0. Then there are a sequenceh n of points inT and a sequencew n of points in Δ withh n ∈(Δ?V h [z 0]) andw n →f 0(z 0) andh n →h 0 andn∞ such that the point shift differentials determined byh n asw n form a Hamilton sequence ofμ 0.展开更多
基金the National Natural Science Foundation of China!(No.19531060), the DoctoralProgram Fundation of the Ministry of Education o
文摘It is proved that if f is a Teichmuller self-mapping of the unit disk with a holomorphic quadratic deferential and satisfies the growth condition m(ψ，r)= o((1 -r)-), r→1, for any s>1, then f is extremal, and there exists a sequence {tn}, 0<tn<1, /lim, tn =1, such that {(tnz)} is a Hamilton sequence. It is the precision of a theorem of Reich-Strebel in 1974, and gives a fairly satisfactory answer to a question of Reich in 1988.
文摘Let S be a Riemann surface with genus p and n punctures. Assume that 3p- 3 ＋ n 〉 0 and n 〉 1. Let a be a puncture of S and let S = S∪{α}. Then all mapping classes in the mapping class group Mods that fixes the puncture a can be projected to mapping classes of Mod$ under the forgetful map. In this paper the author studies the mapping classes in Mods that can be projected to a given hyperbolic mapping class in Mode.
基金supported by the National Natural Science Foundation of China(10971008 and 11371045)
文摘The purpose of this paper is to give a relatively elementary and direct proof of the Delta Inequality, which plays a very important role in the study of the extremal problem of quasiconformal mappings.
基金National Natural Science foundation ofChina!( No.195310 60 ) Doctor Fundsof National Education Committec!( 970 2 4 811)
文摘In 1981, E. Reich established a theorem which stated a sufficient condition for a quasiconformal mapping to be a unique extremal mapping with the given boundary value, and asked whether or not such a uniquely extremal mapping was a Teichmüller mapping in his remark. In this paper, the above open problem was solved and consequently the theorem was strengthened.
文摘The boundary value problem for harmonic maps of the Poincare disc is discussed. The emphasis is on the non-smoothness of the given boundary values in the problem. Let T . be a subspace of the universal Teichmüller space, defined as a set of normalized quasisymmetric homeomorphisms h of the unit circle S onto itself where h admits a quasiconformal extension to the unit disc D with a complex dilatation μ satisfyingwhere ρ(z)|dz|2 is the Poincare metric of D. Let B . be a Banach space consisting of holomorphic quadratic differentials φ in D with normsIt is shown that for any given quasisymmetric homeomorphism h : S1→S1∈ T . , there is a unique quasiconformal harmonic map of D with respect to the Poincare metric whose boundary corresponding is h and the Hopf differential of such a harmonic map belongs to B .
文摘It is well known that certain isotopy classes of oseudo-Anosov maos on a Riemann surface S of non-excluded type can be defined through Dehn twists tα and tβ along simple closed geodesics α and β on S,respectively. Let G be the corresponding Fuchsian group acting on the hyperbolic plane H so that H/G≌S.For any point α∈S,define S = S／{α}.In this article, the author gives explicit parabolic elements of G from which he constructs pseudo-Anosov classes on S that can be projected to a given pseudo-Anosov class on S obtained from Thurston＇s construction.
文摘LET D be the unit disk in the complex plane C and f be a sense preserving quasisymmetrichomeomorphism of D onto itself. Denote by Q a quadrilateral D （z<sub>1</sub>, z<sub>2</sub>, z<sub>3</sub>, z<sub>4</sub> ） with do-main D and vertices z<sub>1</sub>, z<sub>2</sub>, z<sub>3</sub>, z<sub>4</sub> ∈D, and by M（Q） its conformal modulus. We are in-
文摘A constantK 0 (m) (h) is introduced for every quasisymmetric mappingh of the unit circle and every integerm≥4 which contains the constantK 0(h) (indicated by the change in module of the quadrilaterals with vertices on the circle) as a special case. A necessary and sufficient condition is established forK 0 (m) (h) =K 1(h). It is shown that there are infinitely many quasisymmetric mappings of the unit circle having the property thatK 0 (m) (h)<K 1(h), wherek 1(h) is the maximal dilatation ofh.
文摘Two necessary and sufficient conditions for the validity of the conjecture K 0(h)=K 1(h) are given, which are independent of the complex dilatations of extremal quasiconformal mappings, where K 0(h) is the maximal conformal modulus dilatation of the boundary homeomorphism h, K 1(h) is the maximal dilatation of extremal quasiconformal mappings that agree with h on the boundary. In addition, when the complex dilatation of an extremal quasiconformal mapping is known, the proof of the result simplifies Reich and Chen Jixiu-Chen Zhiguo’s result.
基金This work was supported by the National Natural Science Foundation of China(Grant Nos.19901032 and 10171003).
文摘This paper studies the subset of the non-Strebel points in the universal Teichmüller spaceT. Let z0 ∈ Δ be a fixed point. Then we prove that for every non-Strebel pointh, there is a holomorphic curve γ: [0, 1] →T withh as its initial point satisfying the following conditions. (1) The curve γ is on a sphere centered at the base-point ofT, i.e.d T (id, γ(t))=d T (id, h), (t∈[0, 1]). (2) For everyt ∈ (0,1], the variability set Vγ(t)[z0] of γ(t) has non-empty interior, i.e. .
文摘Let QS* (S 1) be the space of quasisymmetric homeomorphisms of the unit circle such that the corresponding subspace of the universal Teichmu¨ller space has Weil-Petersson metric.In this paper we give a necessary condition for a quasisymmetric homeomorphism to belong to QS *(S 1) from the aspect of cross-ratio distortion.
基金Project supported by the National Natural Science Foundation of China (Grant No. 19531060)the Doctoral Education Program Foundation of China
文摘LetT be the universal Teichmüller space viewed as the set of all normalized quasisymmetric homeomorphism of the unit circleS 1=?Δ. Denote byV h [z 0] the variability set ofz 0 with respect toh∈T. The following is proved: Leth 0 be a point ofT. Suppose thatμ 0 is an arbitrarily given extremal Beltrami differential ofh 0 andf 0: μ→μ is a quasiconformal mapping with the Beltrami coefficientμ 0 andf 01s=h 0. Then there are a sequenceh n of points inT and a sequencew n of points in Δ withh n ∈(Δ?V h [z 0]) andw n →f 0(z 0) andh n →h 0 andn∞ such that the point shift differentials determined byh n asw n form a Hamilton sequence ofμ 0.
