Groups of volume-preserving diffeomorphisms of noncompact manifolds and mass flow toward ends

Abstract

Suppose M is a noncompact connected oriented C^infty n-manifold and omega is a positive volume form on M. Let D^+(M) denote the group of orientation preserving diffeomorphisms of M endowed with the compact-open C^infty topology and D(M; omega) denote the subgroup of omega-preserving diffeomorphisms of M. In this paper we propose a unified approach for realization of mass transfer toward ends by diffeomorphisms of M. This argument together with Moser's theorem enables us to deduce two selection theorems for the groups D^+(M) and D(M; omega). The first one is the extension of Moser's theorem to noncompact manifolds, that is, the existence of sections for the orbit maps under the action of D^+(M) on the space of volume forms. This implies that D(M; omega) is a strong deformation retract of the group D^+(M; E^omega_M) consisting of h in D^+(M) which preserves the set E^omega_M of omega-finite ends of M. The second one is related to the mass flow toward ends under volume-preserving diffeomorphisms of M. Let D_{E_M}(M; omega) denote the subgroup consisting of all h in D(M; omega) which fix the ends E_M of M. S.R.Alpern and V.S.Prasad introduced the topological vector space S(M; omega) of end charges of M and the end charge homomorphism c^omega : D_{E_M}(M; omega) to S(M; omega), which measures the mass flow toward ends induced by each h in D_{E_M}(M; omega). We show that the homomorphism c^omega has a continuous section. This induces the factorization D_{E_M}(M; omega) cong ker c^omega times S(M; omega) and it implies that ker c^omega is a strong deformation retract of D_{E_M}(M; omega).