On the $σ_2$-curvature and volume of compact manifolds

Abstract

In this work we are interested in studying deformations of the σ2-curvature and the volume. For closed manifolds, we relate critical points of the total σ2-curvature functional to the σ2-Einstein metrics and, as a consequence of results of H. J. Gursky and J. A. Viaclovsky [25] and Z. Hu and H. Li [27], 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 σ2-curvature which shows that the volume can be controlled by the σ2-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 σ2-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.

Volume comparison theorem with respect to sigma-2 curvature

“…Remark 1.8. When we finished our work, we found the paper [1] also cover the volume comparison theorem with respect to σ 2 -curvature. This paper is organized as follows.…”