By studying the existence problem of locally extremal Beltrami differential posed by F. Gardiner and N. Lakic, we introduce a new definition of the local boundary the same as the usual ones in the case of Jordan domai...By studying the existence problem of locally extremal Beltrami differential posed by F. Gardiner and N. Lakic, we introduce a new definition of the local boundary the same as the usual ones in the case of Jordan domains. Then we show that the answer to the problem of F. Gardiner and N. Lakic is affirmative according to this new definition. Comparing with the usual definitions, the problem of F. Gardiner and N. Lakic is partly answered and the results of G. Cui and Y. Qi is generalized.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 space T. Let z0 ∈△ be a fixed point. Then we prove that for every non-Strebel point h, there is a holomorphic curveγ . [0, ...This paper studies the subset of the non-Strebel points in the universal Teichmüller space T. Let z0 ∈△ be a fixed point. Then we prove that for every non-Strebel point h, there is a holomorphic curveγ . [0, 1] → T with h as its initial point satisfying the following conditions.(1) The curve γ is on a sphere centered at the base-point of T, i.e. dT(id, γ(t)) = dT(id, h), (t ∈ [0, 1]).(2) For every t ∈(0, 1], the variability set Vγ(t)[zo] of γ(t) has non-empty interior, i.e. Vγ(t) [z0] ≠φ.展开更多
基金The author would like to thank Professor Cui Guizhen for helpful discussions.This work was supported by the National Natural Science Foundation of China(Grant No.10271063).
文摘By studying the existence problem of locally extremal Beltrami differential posed by F. Gardiner and N. Lakic, we introduce a new definition of the local boundary the same as the usual ones in the case of Jordan domains. Then we show that the answer to the problem of F. Gardiner and N. Lakic is affirmative according to this new definition. Comparing with the usual definitions, the problem of F. Gardiner and N. Lakic is partly answered and the results of G. Cui and Y. Qi is generalized.
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 space T. Let z0 ∈△ be a fixed point. Then we prove that for every non-Strebel point h, there is a holomorphic curveγ . [0, 1] → T with h as its initial point satisfying the following conditions.(1) The curve γ is on a sphere centered at the base-point of T, i.e. dT(id, γ(t)) = dT(id, h), (t ∈ [0, 1]).(2) For every t ∈(0, 1], the variability set Vγ(t)[zo] of γ(t) has non-empty interior, i.e. Vγ(t) [z0] ≠φ.
