Topology of three-manifolds with positive $P$-scalar curvature

Consider an $n$-dimensional smooth Riemannian manifold $(M^n,g)$ with a given smooth measure $m$ on it. We call such a triple $(M^n,g,m)$ a Riemannian measure space. Perelman introduced a variant of scalar curvature in his recent work on solving Poincaré's conjecture $P(g)=R^m_\infty(g) = R(g) - 2\Delta_g log\phi - \vert\nabla log\phi\vert^2_g$, where $dm = \phi dvol(g)$ and $R$ is the scalar curvature of $(M^n,g)$. In this note, we study the topological obstruction for the $\phi$-stable minimal submanifold with positive $P$-scalar curvature in dimension three under the setting of manifolds with density.