Horizons in noncompact fill-ins of nonnegative scalar curvature

Given a complete Riemannian metric of nonnegative scalar curvature on $\mathrm{\Sigma }\times (-\mathrm{\infty },0]$, where $\mathrm{\Sigma }$ denotes a $2$-sphere, we exhibit conditions that imply the existence of a closed minimal surface homologous to the boundary.