The relationship between Strebel boundary dilatation of a quasisymmetric function h of the unit circle and the dilatation indicated by the change in the modules of the quadrilaterals with vertices on the circle intrig...The relationship between Strebel boundary dilatation of a quasisymmetric function h of the unit circle and the dilatation indicated by the change in the modules of the quadrilaterals with vertices on the circle intrigues many mathematicians. It had been a conjecture for some time that the dilatations Ko（h） and K1（h） of h are equal before Anderson and Hinkkanen disproved this by constructing concrete counterexamples. The independent work of Wu and of Yang completely characterizes the condition for Ko（h） = K1 （h） when h has no substantial boundary point. In this paper, we give a necessary and sufficient condition to determine the equality for h admitting a substantial boundary point.展开更多
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
This paper studies extremal quasiconformal mappings. Some properties of the variability set are obtained and the Hamilton sequences which are induced by point shift differentials are also discussed.
In this paper, we study the boundary dilatation of quasiconformal mappings in the unit disc. By using Strebel mapping by heights theory we show that a degenerating Hamilton sequence is determined by a quasisymmetric f...In this paper, we study the boundary dilatation of quasiconformal mappings in the unit disc. By using Strebel mapping by heights theory we show that a degenerating Hamilton sequence is determined by a quasisymmetric function.展开更多
Let T(S) be a Teichmüller space of a hyperbolic Riemann surface S, viewed as a set of Teichmüller equivalence classes of Beltrami differentials on S. It is shown in this paper that for any extremal Beltrami ...Let T(S) be a Teichmüller space of a hyperbolic Riemann surface S, viewed as a set of Teichmüller equivalence classes of Beltrami differentials on S. It is shown in this paper that for any extremal Beltrami differential μ0 at a given point τ of T(S), there is a Hamilton sequence for μ0 formed by Strebel differentials in a natural way. Especially, such a kind of Hamilton sequence possesses some special properties. As applications, some results on point shift differentials are given.展开更多
Let (?)(z) be holomorphic in the unit disk △ and meromorphic on △. Suppose / is a Teichmuller mapping with complex dilatation In 1968, Sethares conjectured that f is extremal if and only if either (i)(?) has a doubl...Let (?)(z) be holomorphic in the unit disk △ and meromorphic on △. Suppose / is a Teichmuller mapping with complex dilatation In 1968, Sethares conjectured that f is extremal if and only if either (i)(?) has a double pole or (ii)(?) has no pole of order exceeding two on (?)△. The 'if' part of the conjecture had been solved by himself. We will give the affirmative answer to the 'only if' part of the conjecture. In addition, a more general criterion for extremality of quasiconformal mappings is constructed in this paper, which generalizes the 'if' part of Sethares' conjecture and improves the result by Reich and Shapiro in 1990.展开更多
In this paper, we introduce an operator Hμ（z） on L^∞（△） and obtain some of its properties. Some applications of this operator to the extremal problem of quasiconformal mappings are given. In particular, a suffi...In this paper, we introduce an operator Hμ（z） on L^∞（△） and obtain some of its properties. Some applications of this operator to the extremal problem of quasiconformal mappings are given. In particular, a sufficient condition for a point r in the universal Teichmfiller space T（△） to be a Strebel point is obtained.展开更多
Let T0（Δ） be the subset of the universal Teichm¨uller space, which consists all of the elements with boundary dilatation 1. Let SQ（Δ） be the unit ball of the space Q（Δ） of all integrable holomorphic quad...Let T0（Δ） be the subset of the universal Teichm¨uller space, which consists all of the elements with boundary dilatation 1. Let SQ（Δ） be the unit ball of the space Q（Δ） of all integrable holomorphic quadratic differentials on the unit disk Δ and Q0（Δ） be defined as Q0（Δ） = {？ ∈ SQ（Δ） ： there exists a k ∈（0, 1） such that [kˉ？ ｜？｜] ∈ T0（Δ）}. In this paper, we show that Q0（Δ） is dense in SQ（Δ）.展开更多
基金Supported by the National Natural Science Foundation of China(10671174, 10401036)a Foundation for the Author of National Excellent Doctoral Dissertation of China(200518)
文摘The relationship between Strebel boundary dilatation of a quasisymmetric function h of the unit circle and the dilatation indicated by the change in the modules of the quadrilaterals with vertices on the circle intrigues many mathematicians. It had been a conjecture for some time that the dilatations Ko（h） and K1（h） of h are equal before Anderson and Hinkkanen disproved this by constructing concrete counterexamples. The independent work of Wu and of Yang completely characterizes the condition for Ko（h） = K1 （h） when h has no substantial boundary point. In this paper, we give a necessary and sufficient condition to determine the equality for h admitting a substantial boundary point.
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.
基金Project supported by the National Natural Science Foundation of China (No.10171003, No.10231040) the Doctoral Education Program Foundation of China.
文摘This paper studies extremal quasiconformal mappings. Some properties of the variability set are obtained and the Hamilton sequences which are induced by point shift differentials are also discussed.
基金Supported by the National Natural Science Foundation of China （Grant Nos. 10171003 and 10231040） and the Doctoral Education Program Foundation of China
文摘In this paper, we study the boundary dilatation of quasiconformal mappings in the unit disc. By using Strebel mapping by heights theory we show that a degenerating Hamilton sequence is determined by a quasisymmetric function.
基金the National Natural Science Foundation of Chinathe 973 Program Foundation of China.
文摘Let T(S) be a Teichmüller space of a hyperbolic Riemann surface S, viewed as a set of Teichmüller equivalence classes of Beltrami differentials on S. It is shown in this paper that for any extremal Beltrami differential μ0 at a given point τ of T(S), there is a Hamilton sequence for μ0 formed by Strebel differentials in a natural way. Especially, such a kind of Hamilton sequence possesses some special properties. As applications, some results on point shift differentials are given.
文摘Let (?)(z) be holomorphic in the unit disk △ and meromorphic on △. Suppose / is a Teichmuller mapping with complex dilatation In 1968, Sethares conjectured that f is extremal if and only if either (i)(?) has a double pole or (ii)(?) has no pole of order exceeding two on (?)△. The 'if' part of the conjecture had been solved by himself. We will give the affirmative answer to the 'only if' part of the conjecture. In addition, a more general criterion for extremality of quasiconformal mappings is constructed in this paper, which generalizes the 'if' part of Sethares' conjecture and improves the result by Reich and Shapiro in 1990.
基金Supported by the National Science Foundation of China(Grants No.10171003 and 10231040)the Doctoral Education Program Foundation of China
文摘In this paper, we introduce an operator Hμ（z） on L^∞（△） and obtain some of its properties. Some applications of this operator to the extremal problem of quasiconformal mappings are given. In particular, a sufficient condition for a point r in the universal Teichmfiller space T（△） to be a Strebel point is obtained.
基金Supported by National Natural Science Foundation of China(Grant No.11371035)
文摘Let T0（Δ） be the subset of the universal Teichm¨uller space, which consists all of the elements with boundary dilatation 1. Let SQ（Δ） be the unit ball of the space Q（Δ） of all integrable holomorphic quadratic differentials on the unit disk Δ and Q0（Δ） be defined as Q0（Δ） = {？ ∈ SQ（Δ） ： there exists a k ∈（0, 1） such that [kˉ？ ｜？｜] ∈ T0（Δ）}. In this paper, we show that Q0（Δ） is dense in SQ（Δ）.
