紧致流形上交换覆盖的Liouville定理和谱边行为
Liouville theorems and spectral edge behavior on abelian coverings of compact manifolds
摘要：
本文描述了紧致黎曼流形的交换覆盖上的周期椭圆方程（或系统）解的Liouville型定理与该方程在谱边缘的色散关系的结构之间的关系。这里有人说，刘维定理成立，如果空间的解决方案的任何给定的多项式增长是有限维。刘维尔型定理成立的充分必要条件是椭圆算子的实费米面由100个点组成（模倒易格）。因此，这样一个定理一般预期在谱的边缘处成立。多项式增长的解决方案的空间的精确描述取决于一个'均匀化'的常系数运营商确定的色散关系的分析结构。在大多数情况下，简单的显式公式被发现的多项式增长的解决方案的色散曲线方面的空间的尺寸。覆盖物的基底（特别是其尺寸）的作用相当有限，而甲板组是最重要的。结果也建立了超定椭圆系统，特别是导致刘维定理多项式增长全纯函数的交换覆盖紧解析流形。类似的定理适用于紧组合图或量子图的阿贝尔覆盖。
英文摘要：
The paper describes relations between Liouville type theorems for solutions of a periodic elliptic equation (or a system) on an abelian cover of a compact Riemannian manifold and the structure of the dispersion relation for this equation at the edges of the spectrum. Here one says that the Liouville theorem holds if the space of solutions of any given polynomial growth is finite dimensional. The necessary and sufficient condition for a Liouville type theorem to hold is that the real Fermi surface of the elliptic operator consists of finitely many points (modulo the reciprocal lattice). Thus, such a theorem generically is expected to hold at the edges of the spectrum. The precise description of the spaces of polynomially growing solutions depends upon a `homogenized' constant coefficient operator determined by the analytic structure of the dispersion relation. In most cases, simple explicit formulas are found for the dimensions of the spaces of polynomially growing solutions in terms of the dispersion curves. The role of the base of the covering (in particular its dimension) is rather limited, while the deck group is of the most importance. The results are also established for overdetermined elliptic systems, which in particular leads to Liouville theorems for polynomially growing holomorphic functions on abelian coverings of compact analytic manifolds. Analogous theorems hold for abelian coverings of compact combinatorial or quantum graphs.
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