Functions with finite Dirichlet sum of order p and quasi-monomorphisms of infinite graphs

In this paper, we study some potential theoretic properties of connected infinite networks and then investigate the space of p-Dirichlet finite functions on connected infinite graphs, via quasi-monomorphisms. A main result shows that if a connected infinite graph of bounded degrees possesses a quasi-monomorphism into the hyperbolic space form of dimension n and it is not p-parabolic for p > n - 1, then it admits a lot of p-harmonic functions with finite Dirichlet sum of order p.

VARIETIES WITH DEGENERATE GAUSS MAPPINGS IN COMPLEX HYPERBOLIC SPACE FORMS

In analogy with the Gauss mapping for a subvariety in the complex projective space, the Gauss mapping for a subvariety in a complex hyperbolic space form can be defined as a map from the smooth locus of the subvariety to the quotient of a suitable domain in the Grassmannian. For complex hyperbolic space forms of finite volume, it is proved that the Gauss mapping is degenerate if and only if the subvariety is totally geodesic.