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 ${\mathcal D}^+(M)$ denote the group of orientation-preserving diffeomorphisms of $M$ endowed with the compact-open $C^\infty$ topology and let ${\mathcal D}(M; \omega )$ 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 ${\mathcal D}^+(M)$ and ${\mathcal D}(M; \omega )$. 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 ${\mathcal D}^+(M)$ on the space of volume forms. This implies that ${\mathcal D}(M; \omega )$ is a strong deformation retract of the group ${\mathcal D}^+(M; E^\omega _M)$ consisting of $h \in {\mathcal D}^+(M)$, 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 ${\mathcal D}_{E_M}(M; \omega )$ denote the subgroup consisting of all $h \in {\mathcal D}(M; \omega )$ which fix the ends $E_M$ of $M$. S. R. Alpern and V. S. Prasad introduced the topological vector space ${\mathcal S}(M; \omega )$ of end charges of $M$ and the end charge homomorphism $c^\omega : {\mathcal D}_{E_M}(M; \omega ) \to {\mathcal S}(M; \omega )$, which measures the mass flow toward ends induced by each $h \in {\mathcal D}_{E_M}(M; \omega )$. We show that the homomorphism $c^\omega$ has a continuous section. This induces the factorization ${\mathcal D}_{E_M}(M; \omega ) \cong \textrm {ker} c^\omega \times {\mathcal S}(M; \omega )$, and it implies that $\textrm {ker} c^\omega$ is a strong deformation retract of ${\mathcal D}_{E_M}(M; \omega )$.