The aim of this talk is twofold: the topological and geometric classification of complete weighted $H_\phi -$stable surfaces immersed in a manifold with density $(\mathcal{N},g,\phi)$ whose Perelman Scalar curvature, in short, P-Scalar curvature, is nonnegative and the classification of manifolds with density $(\mathcal{N},g,\phi)$ under the existence of a certain compact weighted area-minimizing surface and a lower bound of its $P-$scalar curvature. Here, the P-scalar curvature is defined as $R_\phi^\infty= R - 2 \Delta _g \ln \phi - |\nabla _g \ln \phi |^2$, being $R$ the scalar curvature of $(\mathcal{N},g)$.
