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

Abstract

Suppose M is a noncompact connected oriented C ∞ n-manifold and ω 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 ∞ topology and let D(M; w) denote the subgroup of ω-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;ω). 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; w) is a strong deformation retract of the group D + (M;E ω M ) consisting of h ∈ D + (M), which preserves the set E ω M of ω-finite ends of M. The second one is related to the mass flow toward ends under volume-preserving diffeomorphisms of M. Let D EM (M; w) denote the subgroup consisting of all h ∈ D(M; ω) which fix the ends E M of M. S. R. Alpern and V. S. Prasad introduced the topological vector space S(M; ω) of end charges of M and the end charge homomorphism c ω : D EM (M; ω) → S(M; ω), which measures the mass flow toward ends induced by each h ∈ D EM (M; ω). We show that the homomorphism c ω has a continuous section. This induces the factorization D EM (M; ω) ≅ ker c ω x S(M; ω), and it implies that ker c ω is a strong deformation retract of D EM (M; ω).