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The WKB method for conjugate points in the volumorphism group | | Stephen C. Preston
; | | Date: |
20 Oct 2007 | | Abstract: | In this paper, we are interested in the location of conjugate points along a
geodesic in the volumorphism group of a compact three-dimensional manifold
without boundary (the configuration space of an ideal fluid). As shown in the
author’s previous work, these are typically pathological, i.e., they can occur
in clusters along a geodesic, unlike on finite-dimensional Riemannian
manifolds. (This phenomenon does not occur for the volumorphism groups of
two-dimensional manifolds, which are known to have discrete conjugate points
along any geodesic by Ebin-Misiolek-Preston.) We give an explicit algorithm for
finding them in terms of a certain ordinary differential equation, derived via
the WKB-approximation methods of Lifschitz-Hameiri and Friedlander-Vishik. We
prove that for a typical geodesic in the volumorphism group, there will be
pathological conjugate point locations filling up closed intervals; hence
typically the zeroes of Jacobi fields on the volumorphism group are dense in
intervals. | | Source: | arXiv, 0710.3870 | | Services: | Forum | Review | PDF | Favorites | |
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