Necessary and sufficient conditions for existence of minimal hypersurfaces in spaces of constant curvature

Abstract

1.1. Let (M2, ds 2) be a two-dimensional Riemannian manifold M 2 with metric ds 2 and let p ~ M be a point where the Gaussian curvature K of ds 2 is nonpositive. The Ricci theorem [2, pg. 41 l] states that a necessary and sufficient condition for some neighborhood of p in M to be isometrically and minimally immersed in the euclidean space R 3 is that the metric ~, K ds 2 be flat around p. This condition, usually called the Ricci condition, can be shown to be equivalent to the fact that the metric (- K) ds 2 has constant Gaussian curvature equal to one. The purpose of the present paper is to obtain a generalization of the latter version of the Ricci condition for hypersurfaces of a space M"+l(c) of constant curvature c, c any real number. More precisely, we prove Theor. (1.2) below, for the statement of which we need some notation. We will denote by RicM the quadratic form in a Riemannian manifold (M, ds 2) defined by the Ricci tensor of ds 2. (,) and V will denote the inner product and the Levi-Civitta connection, respectively, of ds 2. We will be considering the case where (n-~ 1)(-RicM+cds 2) is positive definite: then (n- I)(-RicM + cds 2 ) defines a further Riemannian metric on M, and we will denote by ((,)) and ~' the inner product and the Riemannian connect'ion, respectively, of such a metric. Finally, S(M) will denote the bundle over M whose fiber at q ~ M in the space of symmetric linear maps of T~(M).