Prescribing growth type of complete Riemannian manifolds of bounded geometry

Abstract

We describe certain properties of growth types of nondecreasing sequences. We build a complete, connected Riemannian surface of bounded geometry and of a given growth type provided that the type satisfies some natural conditions. 0. Introduction. Growth of leaves plays an important role in the study of topology and dynamics of foliations. The existence of leaves with nonexponential or polynomial growth has some influence on the structure of foliations. Constructing leaves with neither exponential nor polynomial growth can be pretty difficult (see [CC] and references there). The space of all growth types is very rich and contains many types which cannot be compared with polynomial, fractional or exponential ones. In this article, we show that any growth type ξ (not greater than the exponential one and satisfying simple conditions described in Sections 2, 3) can be realized by the volumes of balls on a suitable complete Riemannian manifold of bounded geometry. We believe that in the near future we will be able to apply our construction to obtain leaves of a given growth type on some compact foliated manifolds. 1. Growth types. In this section we recall the notion of the growth type of nondecreasing functions and of complete, connected Riemannian manifolds (compare [HH], [E]). Let I be the set of nonnegative nondecreasing functions on N: I = {f : N→ R+ : f(n) ≤ f(n+ 1) for all n ∈ N}. Define a preorder in I. Let f, h ∈ I. We say that h dominates f (and write f h) if for some A ∈ R+ and B ∈ N, f(n) ≤ Ah(Bn) for any n ∈ N. 2000 Mathematics Subject Classification: Primary 53C99; Secondary 37C85.