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

摘要

Suppose M is a noncompact connected oriented C^infty n-manifold and omega isa positive volume form on M. Let D^+(M) denote the group of orientationpreserving diffeomorphisms of M endowed with the compact-open C^infty topologyand D(M; omega) denote the subgroup of omega-preserving diffeomorphisms of M.In this paper we propose a unified approach for realization of mass transfertoward ends by diffeomorphisms of M. This argument together with Moser'stheorem enables us to deduce two selection theorems for the groups D^+(M) andD(M; omega). The first one is the extension of Moser's theorem to noncompactmanifolds, that is, the existence of sections for the orbit maps under theaction of D^+(M) on the space of volume forms. This implies that D(M; omega) isa strong deformation retract of the group D^+(M; E^omega_M) consisting of h inD^+(M) which preserves the set E^omega_M of omega-finite ends of M. The secondone is related to the mass flow toward ends under volume-preservingdiffeomorphisms of M. Let D_{E_M}(M; omega) denote the subgroup consisting ofall h in D(M; omega) which fix the ends E_M of M. S.R.Alpern and V.S.Prasadintroduced the topological vector space S(M; omega) of end charges of M and theend charge homomorphism c^omega : D_{E_M}(M; omega) to S(M; omega), whichmeasures the mass flow toward ends induced by each h in D_{E_M}(M; omega). Weshow that the homomorphism c^omega has a continuous section. This induces thefactorization D_{E_M}(M; omega) cong ker c^omega times S(M; omega) and itimplies that ker c^omega is a strong deformation retract of D_{E_M}(M; omega).