Fredholm properties of Riemannian exponential maps on diffeomorphism groups
成果类型：
Article
署名作者：
Misiolek, Gerard; Preston, Stephen C.
署名单位：
University of Colorado System; University of Colorado Boulder; University of Notre Dame
刊物名称：
INVENTIONES MATHEMATICAE
ISSN/ISSBN：
0020-9910
DOI：
10.1007/s00222-009-0217-3
发表日期：
2010
页码：
191-227
关键词：
摘要：
We prove that exponential maps of right-invariant Sobolev H (r) metrics on a variety of diffeomorphism groups of compact manifolds are nonlinear Fredholm maps of index zero as long as r is sufficiently large. This generalizes the result of Ebin et al. (Geom. Funct. Anal. 16, 2006) for the L (2) metric on the group of volume-preserving diffeomorphisms important in hydrodynamics. In particular, our results apply to many other equations of interest in mathematical physics. We also prove an infinite-dimensional Morse Index Theorem, settling a question raised by Arnold and Khesin (Topological methods in hydrodynamics. Springer, New York, 1998) on stable perturbations of flows in hydrodynamics. Finally, we include some applications to the global geometry of diffeomorphism groups.
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