The WKB method for conjugate points in the volumorphism group

TL;DR

This paper develops an explicit WKB-based algorithm to locate conjugate points along geodesics in the volumorphism group of 3D manifolds, revealing their typically dense and clustered nature unlike in lower dimensions.

Contribution

It introduces a novel WKB approximation method for identifying conjugate points in infinite-dimensional groups, highlighting their complex clustering behavior.

Findings

Conjugate points can occur in dense clusters along geodesics.

The algorithm explicitly finds conjugate points via an ODE derived from WKB methods.

Pathological conjugate points fill entire intervals, unlike in finite-dimensional cases.

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.