Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2013; 50(4): 1201-1207

Printed July 1, 2013

https://doi.org/10.4134/BKMS.2013.50.4.1201

Copyright © The Korean Mathematical Society.

Structure of stable minimal hypersurfaces in a Riemannian manifold of nonnegative Ricci curvature

Jeong-Jin Kim and Gabjin Yun

Myong Ji University, Myong Ji University

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 $\int_M |A|^2\, dv = \infty$, 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 \cite{l-w} 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 \cite{c-s-z} as a corollary.

Keywords: stable minimal hypersurface, end, $L^2$ harmonic form, parabolicity, non-parabolicity

MSC numbers: 53C21