On the $\sigma_2$-curvature and volume of compact manifolds

In this work we are interested in studying deformations of the $\sigma_2$-curvature and the volume. For closed manifolds, we relate critical points of the total $\sigma_2$-curvature functional to the $\sigma_2$-Einstein metrics and, as a consequence of results of H. J. Gursky and J. A. Viaclovsky (2001) and Z. Hu and H. Li (2004), we obtain a sufficient and necessary condition for a critical metric to be Einstein. Moreover, we show a volume comparison result for Einstein manifolds with respect to $\sigma_2$-curvature which shows that the volume can be controlled by the $\sigma_2$-curvature under certain conditions. Next, for compact manifold with nonempty boundary, we study variational properties of the volume functional restricted to the space of metrics with constant $\sigma_2$-curvature and with fixed induced metric on the boundary. We characterize the critical points to this functional as the solutions of an equation and show that in space forms they are geodesic balls. Studying second order properties of the volume functional we show that there is a variation for which geodesic balls are indeed local minimum in a natural direction.