Jürgen Eichhorn

The manifold structure of maps between open manifolds

We establish in a canonical manner a manifold structure for the completed space of bounded maps between open manifoldsM andN, assuming thatM andN are endowed with Riemannian metrics of bounded geometry up to a certain order. The identity component of the corresponding diffeomorphisms is a Banach manifold and metrizable topological group.

GAUGE THEORY ON OPEN MANIFOLDS OF BOUNDED GEOMETRY

On compact manifolds (Mn, g) and for r>n/2+1 the configuration space is a well-defined object. is an affine space with a Sobolev space as vector space, and a Hilbert Lie group which acts smoothly and properly on . is a stratified space with Hilbert manifolds as strata. The existence problem has been solved for many interesting cases by Cliff Taubes and the description of the moduli space of instantons has been given by Donaldson. On noncompact manifolds none of the approaches of the compact case is further valid. We present here an intrinsic, self-consistent approach for gauge theory on open manifolds of bounded geometry up to order n/2+2. The main idea is to endow the space CP of gauge potentials and the gauge group with an intrinsic Sobolev topology. Bounded geometry of the underlying manifold and the considered connections provides all the Sobolev theorems which are needed to prove the existence of instantons if G=SU(2). We prove the existence of instantons if (M4, g) satisfies a certain spectral condition and has a positive definite L2 intersection form.

Spaces of Riemannian metrics on open manifolds

We define for the set M of metrics on an open manifold Mn suitable uniform structures, obtain completed spaces b,mM or Mr(I, Bk), respectively and calculate for each component of Mr(I, Bk ) the infinitedimensional geometry. In particular, we show that the sectional curvature is non positive.

Form preserving diffeomorphisms on open manifolds

We prove that on open manifolds of bounded geometry satisfying a certain spectral condition the component of the identity Dinfw,0supr of form preserving diffeomorphisms is a submanifold of the identity component of all bounded Sobolev diffeomorphisms. Dinfw,0supr inherits a natural Riemannian geometry and we can solve Euler equations in this context.

A priori estimates in geometry and Sobolev spaces on open manifolds

Introduction. For bounded domains inR 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, 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.

The configuration space of gauge theory on open manifolds of bounded geometry

We define suitable Sobolev topologies on the space CP (Bk, f) of connections of bounded geometry and finite Yang–Mills action and the gauge group and show that the corresponding configuration space is a stratified space. The underlying open manifold is assumed to have bounded geometry.

Seiberg-Witten Theory

We give an introduction into and exposition of Seiberg-Witten theory.