Peter Kuchment, Yehuda Pinchover
Liouville theorems and spectral edge behavior on abelian
coverings of compact manifolds
(138K, LATEX)
ABSTRACT. 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.