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

Author: Tatsuhiko Yagasaki
Journal: Trans. Amer. Math. Soc. 362 (2010), 5745-5770
MSC (2010): Primary 57S05, 58D05
Published electronically: June 17, 2010
MathSciNet review: 2661495
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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 {\rm ker} c^\omega \times {\mathcal S}(M; \omega)$, and it implies that $ {\rm ker} c^\omega$ is a strong deformation retract of $ {\mathcal D}_{E_M}(M; \omega)$.

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Additional Information

Tatsuhiko Yagasaki
Affiliation: Division of Mathematics, Graduate School of Science and Technology, Kyoto Institute of Technology, Kyoto, 606-8585, Japan

Keywords: Group of volume-preserving diffeomorphisms, mass flow, end charge homomorphism, $\sigma$-compact manifold
Received by editor(s): June 9, 2008
Published electronically: June 17, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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