Almost Invariant Submanifolds for Compact Group Actions

A compact (not necessarily connected) Lie group G carries a (unique) biinvariant probability measure. Using this measure, one can average orbits of actions of G on affine convex sets to obtain fixed points. In particular, if G acts on a manifoldM , G leaves invariant a riemannian metric onM , and this metric can sometimes be used to obtain fixed points for the nonlinear action of G on M itself. By this method, Élie Cartan proved that a compact group G acting by isometries on a simply-connected manifold M of nonpositive sectional curvature always has a fixed point.1 Cartan’s result was extended by Grove and Karcher [14] to arbitrary manifolds under the assumption that the G-action has an orbit which is sufficiently small relative to a distance scale provided by the geometry of M . In this paper, we develop a method for averaging nearby submanifolds in a riemannian manifold. This enables us to extend the Grove-Karcher theorem from points to submanifolds; i.e. we establish that, if a riemannian Research partially supported by NSF Grants DMS-96-25122 and DMS-99-71505 and the Miller Institute for Basic Research in Science. In Note III, n 19 of [8], Cartan proved the fixed point theorem for a finite group; he then remarked on p. 19 of [7] that the same argument works for a compact group. See Theorem 13.5 in Chapter 1 of Helgason’s book [16] for a full proof in the compact case.