几何函数论：拉回因式分解的艺术
Geometric function theory: the art of pullback factorization
摘要：
本文发展了一般度量测度空间中拟正则映射理论的基础。特别地，当度量测度空间具有局部有界几何时，证明了具有局部有界重数的离散开映射的9种拟正则性定义在数量上是等价的.我们还表明，这些影响中的一些，虽然不是全部，仍然是真实的相当一般的假设。主要的新工具出现在我们的方法是一个强大的因式分解，称为拉回因式分解，拟正则映射到1-BLD映射和拟共形映射在局部紧完备度量测度空间的组成。这种分解也为欧几里得空间中的拟正则映射理论带来了新的观点，特别是引入了拟对称映射的分支映射，并证明了它与拟正则映射在定量上是局部等价的.作为这些新技巧的应用，我们回答了这一领域的一些著名的公开问题，并刻画了具有局部有界几何的度量空间中的BLD映射。特别地，Heinonen-Rickman的一个猜想被证明是错误的，Heinonen-Rickman的一个公开问题得到了肯定的解决。
英文摘要：
In this paper, we develop the foundations of the theory of quasiregular mappings in general metric measure spaces. In particular, nine definitions of quasiregularity for a discrete open mapping with locally bounded multiplicity are proved to be quantitatively equivalent when the metric measure spaces have locally bounded geometry. We also demonstrate that some, though not all, of these implications remain true under fairly general hypotheses. The major new tool appeared in our approach is a powerful factorization, termed the pullback factorization, of a quasiregular mapping into the composition of a 1-BLD mapping and a quasiconformal mapping in locally compact complete metric measure spaces. This factorization also brings fundamental new point of view of the theory of quasiregular mappings in Euclidean spaces, in particular, a branched counterpart of quasisymmetric mappings is introduced and is shown to be locally equivalent with quasiregular mappings, quantitatively. As applications of our new techniques, we answer some well-known open problems in this field and characterize BLD mappings in metric spaces with locally bounded geometry. In particular, a conjecture of Heinonen--Rickman is shown to be false and an open problem of Heinonen--Rickman gets solved affirmatively.
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