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The structure of \({\phi}\)-stable minimal hypersurfaces in manifolds of nonnegative P-scalar curvature

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Abstract

Suppose (N n, g) is an n-dimensional Riemannian manifold with a given smooth measure m. The P-scalar curvature is defined as \({P(g)=R^m_\infty(g)=R(g)-2\Delta_g{\rm log}\,\phi-|\nabla_g{\rm log}\,\phi|_g^2}\), where \({dm=\phi\,dvol(g)}\) and R(g) is the scalar curvature of (N n, g). In this paper, under a technical assumption on \({\phi}\), we prove that \({\phi}\)-stable minimal oriented hypersurface in the three-dimensional manifold with nonnegative P-scalar curvature must be conformally equivalent to either the complex plane \({\mathbb{C}}\) or the cylinder \({\mathbb{R}\times\mathbb{S}^1}\).

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  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907-2067, USA

    Pak Tung Ho

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  1. Pak Tung Ho
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Correspondence to Pak Tung Ho.

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Ho, P.T. The structure of \({\phi}\)-stable minimal hypersurfaces in manifolds of nonnegative P-scalar curvature. Math. Ann. 348, 319–332 (2010). https://doi.org/10.1007/s00208-010-0482-x

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  • DOI: https://doi.org/10.1007/s00208-010-0482-x

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