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STRUCTURE OF STABLE MINIMAL HYPERSURFACES IN A RIEMANNIAN MANIFOLD OF NONNEGATIVE RICCI CURVATURE

  • Received : 2012.08.03
  • Published : 2013.07.31

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 ${\int}_M{\mid}A{\mid}^2\;dv={\infty}$, 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 $L^2$ 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.

Keywords

stable minimal hypersurface;end;$L^2$ harmonic form;parabolicity;non-parabolicity

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Acknowledgement

Supported by : National Research Foundation of Korea(NRF)