Minimal orbits of metrics
In the wonderful paper by \textit{W.-Y. Hsiang} [Proc. Natl. Acad. Sci. USA 56, 5-6 (1966; Zbl 0178.55904)] it was proved that for an isometric action of a Lie group on a finite dimensional Riemannian manifold \(X\) those orbits, that have minimal volume among all nearby orbits of the same type, are in fact minimal submanifolds of \(X\). In the infinite dimensional context, where \(X\) is the space of connections on a principal bundle over a compact manifold \(M\), and the Lie group is the gauge group, minimal orbits have recently been defined and examples given using a zeta function regularization. In the present paper, the authors treat a different infinite dimensional case, namely the isometric action of the group of diffeomorphisms of a compact manifold on the space of Riemannian metrics on the manifold. They define minimal orbits for this action by a zeta function regularization and show that odd dimensional isotropy irreducible homogeneous spaces give rise to minimal orbits, thereby providing the first known examples of minimal submanifolds of infinite dimension and codimension. They also find a flat 2-torus giving a stable minimal orbit, and they prove that an orbit of isolated diffeomorphism type is minimal, as in the finite dimensional case.
scientific article; zbMATH DE number 1112286
scientific article