拟共形映射中的极值问题

The extremal problems of quasiconformal mappings are mainly concerned in this project. The problem of whether the maximal dilatation is equal to the extremal maximal dilatation of a quasisymmetric mapping with substantial points is studied. We prove that the two.quantities are equal for a large class of such mappings. Together with a result obtained later by Shen YuLian, a problem on quasiconformal mappings is answered. Some related properties of variability sets of quasisymmtric mappings and the Hamilton sequences formed by point.shift differentials are studied. We prove that a Hamilton sequence formed by point shift differentials is either convergent in norm or is a commonHamilton sequence for all extremal Beltrami differentials of the same class. Using variability sets, the non-Strebel points are divided into two class and studied. We prove that every non-Strebel point is the end point.of a holomorphic arc, on the same sphere with such point and centered at the base point, on which all points (may not include such end point) have variability sets with non-empty interiors. The existence problem of locally extremal Beltrami differentials, which is closely related to the substantial points and common Hamilton sequences of extremal Beltrami differentials and which is of its own independent meaning, are also studied. We prove that the locally extremal Beltrami differentials always exist when the domain is a disk or the boundary of the domain is not too bad. This partly and affirmatively answers an open problem posed by F. Gardiner and N. Lakic.

主要研究Strebel提出的变化集的结果,尤其是其边界点的结构;研究拓扑四边形的模与T(A)蟹荢trebel点的极值最大伸缩商的关系;研究不唯一极值的同伦类中Teichmuller极值存在性与唯一性.通过这些问题的研究势必导致对两个由来已久的极值问题,即极值映射的划画与唯一性问题产生更深入的了解,并且对为Teichmuller空间中Strebel点与非Strebel点的研究产生影响..