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

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 $${\mathbb{R}\times\mathbb{S}^1}$$ .