The structure of {\phi}-stable minimal hypersurfaces in manifolds of nonnegative P-scalar curvature |
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Authors: | Pak Tung Ho |
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Institution: | 1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907-2067, USA
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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}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)=Rm¥(g)=R(g)-2Dglog f-|?glog f|g2{P(g)=R^m_\infty(g)=R(g)-2\Delta_g{\rm log}\,\phi-|\nabla_g{\rm log}\,\phi|_g^2}, where dm=f dvol(g){dm=\phi\,dvol(g)} and R(g) is the scalar curvature of (N
n
, g). In this paper, under a technical assumption on f{\phi}, we prove that f{\phi}-stable minimal oriented hypersurface in the three-dimensional manifold with nonnegative P-scalar curvature must be conformally equivalent to either the complex plane
\mathbbC{\mathbb{C}} or the cylinder
\mathbbR×\mathbbS1{\mathbb{R}\times\mathbb{S}^1}. |
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