Math. Z. 190, 73-82 (1985) Mathematische Zeitschrift 9 Springer-Verlag 1985 Minimal Hypersurfaces of a Positive Scalar Curvature Manifold Sebastiio Almeida* Departamento de Matemfitica, Universidade Federal do Cear/t Campus do Pici, 60000 Fortaleca-CE, Brazil wO. Introduction This paper deals with the geometry of manifolds of positive scalar curvature. In w1 we prove a basic theorem concerning the existence of positive scalar curvature on the double of a riemannian manifold X. We show that if the sca- lar curvature of X is > l and the mean curvature of its boundary with respect to the interior normal is > 0, then the double of X also carries a metric of pos- itive scalar curvature. In w we use the basic theorem to construct a series of examples of im- mersed hypersurfaces in the n-sphere S" which can never be minimal for any metric of positive scalar curvature on S n. We recall that by the Phillips' Submersion Theorem, [8], an open n-dimen- sional manifold can be submersed in IRn if and only if it is parallelizable. Let Mn+l= T,+ l\/fi=+ 1 be a torus minus an open (n + 1)-ball. By the Phillips' theo- rem we know there is an immersion f: M"+I~S ~+~. We claim that the restric- tion of f to the boundary gives an immersion f: S"~S "+1 which can never be minimal for any metric of positive scalar curvature on S "+ 1. In fact the double of M is T n+l#T "+~ and this does not support a metric of positive scalar curvature (cf. [3]). More generally, we let M be a compact (n+ 1)-dimensional manifold with boundary 0M, such that its double carries no metric of positive scalar curvature. An example of such a manifold is X#T "+~ where X is any parallelizable manifold. We prove the following result. Theorem2.7. If M is parallelizable and f: M ~ N is an immersion of M onto a (n +1)-dimensional manifold N, then fit?M: OMeN cannot have mean curvature >= 0 for any metric of positive scalar curvature on N. For the special case dim M=3, the condition of being parallelizable is automatically satisfied for any orientable 3-manifold M. In [3], Gromov-Law- son proved that if a compact orientable 3-manifold M has a K(~, 1) factor in * Supported by CNPq, and Universidade Federal do Cearfi- Brasil