The concept and criteria of the capacitive type of the non-compact riemannian manifold based on the generalized capacity

Let be a non-compact 𝑛-dimensional Riemannian manifold and let > 1 be a fixed real number. We call (𝐹, 𝑝)-capacity of a compact set ⊂ a value inf ∫︀ 𝑀𝑛(𝐹(𝑥,∇𝑢))𝑝𝑑𝑣, where the exact lower bound is taken over all smooth functions finite in and such that ≥ 1 on 𝐾. Function = 𝐹(𝑥, ), (𝑥, ) ∈ is smooth, non-negative and satisfies certain general conditions. A special case of (𝐹, 𝑝)-capacity is, e. g., the conformal capacity when 𝐹(𝑥, ) = | | and = 𝑛. We based this notion of (𝐹, 𝑝)-capacity on the work of G. Choquet, V.G. Mazya, and V.M. Miklyukov. Let us introduce the concept of the type of a non-compact manifold as follows. We say that is of (𝐹, 𝑝)-parabolic type, if the (𝐹, 𝑝)-capacity of some non-degenerate compact ⊂ is zero. Otherwise, we say that manifold is of (𝐹, 𝑝)-hyperbolic type. Like in the classical case, this notion of (𝐹, 𝑝)-type of the non-compact Riemannian manifold is invariant with respect to the specific choice of the compact set 𝐾. We prove the criteria for the manifold to be of (𝐹, 𝑝)-parabolic or (𝐹, 𝑝)- hyperbolic type. Special cases of these are the well-known criteria of conformal type of a Riemannian manifold expressed in terms of growth of the volume (𝑟) of geodesic balls or of area 𝑆(𝑟) of their boundary spheres of radius 𝑟. In the general case of criteria of (𝐹, 𝑝)-type of manifold the role of the class of complete metrics conformal to the initial metric of the manifold takes on the class of exhaustion functions ℎ of manifold 𝑀𝑛, and the roles of (𝑟) and 𝑆(𝑟) are taken by functions 𝑉𝐹,𝑝,ℎ(𝑟) = ∫︀ ℎ≤𝑟(𝐹(𝑥,∇ℎ))𝑝𝑑𝑣 and 𝑆𝐹,𝑝,ℎ(𝑟) = ∫︀ ℎ=𝑟(𝐹(𝑥,∇ℎ))𝑝(𝑑 /|∇ℎ|), respectively. The criteria themselves are expressed in terms of the growth of these functions. For instance, the following conditions ∫︁ +∞ (︂ 𝑉𝐹,𝑝,ℎ(𝑟) )︂ 1 𝑝-1 = ∞, ∫︁ +∞ (︂ 1 𝑆𝐹,𝑝,ℎ(𝑟) )︂ 1 𝑝-1 = ∞ characterize the (𝐹, 𝑝)-parabolic type of the non-compact Riemannian manifold.