非紧流形的保体积自同态群与质量流
Groups of volume-preserving diffeomorphisms of noncompact manifolds and mass flow toward ends
摘要：
设M是一个非紧的连通定向C^infty n-流形，Ω是M上的一个正体积形式。设D^+（M）表示M的具有紧开C^infty拓扑的保向同态群，D（M; omega）表示M的保omega同态子群.本文提出了用M的双同态实现向端传质的统一方法。这个论证与莫泽定理一起使我们能够推导出群D^+（M）和D（M; ω）的两个选择定理。第一个是将Moser定理推广到非紧流形上，即在体积形式空间上D^+（M）作用下的轨道映射的截面的存在性。这意味着D（M; Ω）是由D^+（M）中的h组成的群D^+（M; E^omega_M）的强形变收缩核，它保持M的Ω有限端的集合E^omega_M。第二个问题与M的保体积同态下的质量流有关。设D_{E_M}（M; ω）表示由D（M; ω）中固定M的端点E_M的所有h组成的子群。S.R.Alpern和V.S.Prasad引入了M的端电荷的拓扑向量空间S（M; omega）和端电荷同态c^omega：D_{E_M}（M; omega）到S（M; omega），它测量由D_{E_M}（M; omega）中的每个h引起的朝向端的质量流。我们证明了同态c^omega有一个连续截面。这导致因子分解D_{E_M}（M; omega）cong ker c^omega乘以S（M; omega），并且它意味着ker c^omega是D_{E_M}（M; omega）的强形变收缩核。
英文摘要：
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).
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