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Complete harmonic stable minimal hypersurfaces in a Riemannian manifold

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Abstract.

In this paper, we will introduce the notion of harmonic stability for complete minimal hypersurfaces in a complete Riemannian manifold. The first result we prove, is that a complete harmonic stable minimal surface in a Riemannian manifold with non-negative Ricci curvature is conformally equivalent to either a plane R 2 or a cylinder R × S 1, which generalizes a theorem due to Fischer-Colbrie and Schoen [12].

The second one is that an n ≥ 2-dimensional, complete harmonic stable minimal, hypersurface M in a complete Riemannian manifold with non-negative sectional curvature has only one end if M is non-parabolic. The third one, which we prove, is that there exist no non-trivial L 2-harmonic one forms on a complete harmonic stable minimal hypersurface in a complete Riemannian manifold with non-negative sectional curvature. Since the harmonic stability is weaker than stability, we obtain a generalization of a theorem due to Miyaoka [20] and Palmer [21].

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Authors and Affiliations

  1. Saga University, Saga, Japan

    Qing-Ming Cheng

  2. Kyungpook National University, Taegu, South Korea

    Young Jin Suh

Authors
  1. Qing-Ming Cheng
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  2. Young Jin Suh
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Additional information

Research partially Supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology, Japan.

The author’s research was supported by grant Proj. No. KRF-2007-313-C00058 from Korea Research Foundation, Korea.

Authors’ addresses: Qing-Ming Cheng, Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga 840-8502, Japan; Young Jin Suh, Department of Mathematics, Kyungpook National University, Taegu 702-701, South Korea

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Cheng, QM., Suh, Y. Complete harmonic stable minimal hypersurfaces in a Riemannian manifold. Monatsh Math 154, 121–134 (2008). https://doi.org/10.1007/s00605-007-0508-y

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  • DOI: https://doi.org/10.1007/s00605-007-0508-y

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