发表机构
Northern Illinois University（北伊利诺伊大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究有理映射与一致拟正则映射的Sobolev混合估计一致性，通过中心迹常数刻画模空间紧性，并给出谱结构及紧性判据。
AI 中文摘要
我们研究有理映射和一致拟正则映射的Sobolev混合估计的一致性。对于固定次数的有理映射，最优中心Sobolev迹常数$A_f$是Möbius模空间上的连续正常函数。因此，$A_f$的一致有界性刻画了模空间中的相对紧性，并且在每个次数下都存在极小化类。在闭$n$流形上具有不变共形结构的次数$d\ge2$的一致拟正则自同态的设定中，从临界Sobolev能量到平衡测度的$L^1$的第$k$个中心转移算子的范数为$A_f d^{-k/n}$。在零均值Sobolev空间上，谱和Fredholm本质谱是半径为$d^{-1/n}$的闭圆盘，其内部处处具有无穷维特征空间。证明使用了拉回的能标度、有界平衡迹以及来自中等质量原子的障碍。结合共形重心归一化和DeMarco--Faber退化定理，该障碍给出了模紧性判据。显式族说明了由坐标变换引起的退化与集中之间的区别。
英文摘要
We study uniformity of Sobolev mixing estimates for rational maps and uniformly quasiregular mappings. For rational maps of fixed degree, the optimal centered Sobolev trace constant $A_f$ is a continuous proper function on Möbius moduli space. Uniform bounds on $A_f$ therefore characterize relative compactness in moduli, and a minimizing class exists in every degree. In the setting of uniformly quasiregular endomorphisms of degree $d\ge2$ on closed $n$-manifolds with an invariant conformal structure, the $k$th centered transfer operator from critical Sobolev energy into $L^1$ of the equilibrium measure has norm $A_f d^{-k/n}$. On the mean-zero Sobolev space, the spectrum and Fredholm essential spectrum are the closed disk of radius $d^{-1/n}$, with infinite-dimensional eigenspaces throughout its interior. The proofs use the energy scaling of pullback, a bounded equilibrium trace, and an obstruction from atoms of intermediate mass. Combined with conformal barycenter normalization and DeMarco--Faber's degeneration theorem, this obstruction gives the moduli compactness criterion. Explicit families illustrate the distinction between degeneration and concentration caused by changes of coordinates.
