Structure of stable minimal hypersurfaces in a Riemannian manifold of nonnegative Ricci curvature
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Let N be a complete Riemannian manifold with nonnegative Ricci curvature and let M be a complete noncompact oriented stable minimal hypersurface in N. We prove that if M has at least two ends and RM |A|2 dv = ∞, then M admits a nonconstant harmonic function with finite Dirichlet integral, where A is the second fundamental form of M. We also show that the space of L2 harmonic 1-forms on such a stable minimal hypersurface is not trivial. Our result is a generalization of one of main results in [12] because if N has nonnegative sectional curvature, then M admits no nonconstant harmonic functions with finite Dirichlet integral. And our result recovers a main theorem in [3] as a corollary.
Abstract
Let N be a complete Riemannian manifold with nonnegative Ricci curvature and let M be a complete noncompact oriented stable minimal hypersurface in N. We prove that if M has at least two ends and RM |A|2 dv = ∞, then M admits a nonconstant harmonic function with finite Dirichlet integral, where A is the second fundamental form of M. We also show that the space of L2 harmonic 1-forms on such a stable minimal hypersurface is not trivial. Our result is a generalization of one of main results in [12] because if N has nonnegative sectional curvature, then M admits no nonconstant harmonic functions with finite Dirichlet integral. And our result recovers a main theorem in [3] as a corollary.
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