Local pointwise estimates for solutions of the σ2 curvature equation on 4-manifolds

Abstract

The study of the kth elementary symmetric function of the Weyl-Schouten curvature tensor of a Riemannian metric, the so called σ k curvature, has produced many fruitful results in conformal geometry in recent years, especially when the dimension of the underlying manifold is 3 or 4. In these studies in conformal geometry, the deforming conformal factor is considered to be a solution of a fully nonlinear elliptic PDE. Important advances have been made in recent years in the understanding of the analytic behavior of solutions of the PDE, including the adaptation of Bernstein-type estimates in integral form, global and local derivative estimates, classification of entire solutions, and analysis of blowing-up solutions. Most of these results require derivative bounds on the σ k curvature. The derivative estimates also require an a priori L ∞ bound on the solution. This work provides local L ∞ and Harnack estimates for solutions of the σ 2 bounds on the σ 2 curvature, and the natural assumption of small volume (or total σ 2 curvature).