Analysis Seminar

Consider the flow of ideal incompressible fluid in a bounded 2-d domain $M$ (say, $M= 3DT^2$, the 2-d torus). In the Lagrange formulation, the flow is a geodesic $f_t$ on the group $SDif f(M)$ of volume-preserving diffeomorphisms of $M$ with respect to the right-invariant $L^2$-metric. The 2-point problem is formulated as follows:given two diffeomorphisms $g$ and $h$, find a geodesic $f_t$, $0 \leq t \leq 1$ such that $f_0= 3Dg$, $f_1= 3Dh$.

THEOREM: The 2-point problem has a solution for any $g, h \in SDif f(M)$.

(Note that the geodesic whose existence is claimed in this theorem is not always the shortest curve connecting $g$ and $h$; the latter can not exist at all.) This result reminds the Hopf-Rinow theorem; however, they have little in common, because the infinite-dimensional manifold $SDif f(M)$ is neither locally-compact nor complete in the $L^2$-metric. The proof is based on completely different ideas on the geometrical structure of the diffeomorphism group and the exponential map thereon; one of the by-products is an alternative proof of the original Hopf-Rinow theorem.