A priori estimates in geometry and Sobolev spaces on open manifolds

Introduction. For bounded domains in $R^n$ satisfying the cone condition there are many embedding and module structure theorem for Sobolev spaces which are of great importance in solving partial differential equations. Unfortunately, most of them are wrong on arbitrary unbounded domains or on open manifolds. On the other hand, just these theorems play a decisive role in foundations of nonlinear analysis on open manifolds and in solving partial differential equations. This was pointed out by the author in particular in [4]. But if the open Riemannian manifold $(M^n,g)$ and the considered Riemannian vector bundle (E,h) → M have bounded geometry of sufficiently high order then most of the Sobolev theorems can be preserved. The key for this are a priori estimates for the connection coefficients and the exponential map coming from curvature bounds. By means of uniform charts and trivializations and a uniform decomposition of unity the local euclidean arguments remain applicable. Only the compactness of embeddings is no more valid. This is the content of our main section 4.