GROUPS OF VOLUME-PRESERVING DIFFEOMORPHISMS OF NONCOMPACT MANIFOLDS AND MASS FLOW TOWARD ENDS

摘要

Suppose M is a noncompact connected oriented C~∞ n-manifoldand 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; ω) denote the subgroup of ω-preserving diffeomorphismsof 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 ; ω) is a strong deformation retract of the group D~+(M; E_MM~ω ) consisting of h ∈D~+(M), which preserves the set E_M~ω of ω- finite ends of M. The second one is related to the mass flow toward ends under olumepreserving diffeomorphisms of M. Let D_(E_M) (M;ω) 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_(E_M) (M; ω)→S(M; ω), which measures the mass flow toward ends induced by each h ∈ D_(E_M) (M; ω). We show that the homomorphism c~ω has a continuous section. This induces the factorization D_(E_M) (M;ω) ≈ker c~ω × S (M ;ω), and it implies that ker c~ω is a strong deformation retract of D_(E_M) (M; ω).