The WKB method for conjugate points in the volumorphism group
Preston, Stephen C.
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.
Mathematics - Analysis of PDEs, Mathematics - Differential Geometry