On the structure of complete hypersurfaces in a Riemannian manifold of nonnegative curvature and L 2 harmonic forms

Abstract

Let M n be a complete oriented noncompact hypersurface in a complete Riemannian manifold N n+1 of nonnegative sectional curvature with $${2 \leq n \leq 5}$$ . We prove that if M satisfies a stability condition, then there are no non-trivial L 2 harmonic one-forms on M. This result is a generalization of a well-known fact in the case when M is a stable minimally immersed hypersurface. As a consequence, we show that if the mean curvature of M is constant, then either M must have only one end or M splits into a product of $${\mathbb{R}}$$ and a compact manifold with nonnegative sectional curvature. In case $${n \geq 5}$$ , we also show that the same result holds if the absolute value of the mean curvature is less than or equal to the ratio of the norm of the second fundamental form to the dimension of a hypersurface.