Foliation by constant mean curvature spheres

Rugang Ye

Vol. 147 (1991), No. 2, 381–396

DOI: 10.2140/pjm.1991.147.381

Abstract

Let M be a Riemannian manifold of dimension n + 1 and p ∈ M. Geodesic spheres around p of small radius constitute a smooth foliation. We shall show that this foliation can be perturbed into a foliation whose leaves are spheres of constant mean curvature, provided that p is a nondegenerate critical point of the scalar curvature function of M. The obtained foliation is actually the unique foliation by constant mean curvature hypersurfaces which is regularly centered at p (Definition 1.1). On the other hand, if p is not a critical point of the scalar curvature function, then there exists no such foliation.

Mathematical Subject Classification 2000

Primary: 53C12

Milestones

Received: 18 April 1989

Published: 1 February 1991