2004/06/02 by Zheng-Chao Han, Han, Zheng-Chao
Computer Science · Mathematics · #35B33 #35B45 (secondary) #35J60 #58J05 (primary) #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #math.AP #math.DG #msc:35B33 #msc:35B45 #msc:35J60 #msc:58J05
paper · pdf · doi:10.48550/arxiv.math/0406027
arxiv created 2004/06/02 · openalex publication_date 2004/06/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The study of the k-th elementary symmetric function of the Weyl-Schouten curvature tensor of a Riemannian metric, the so called σk 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, 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 solutoins. Most of these results require derivative bounds on the σk 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 σ2 curvature equation on 4 manifolds, under only Lp bounds on the σ2 curvature, and the natural assumption of small volume(or total σ2 curvature).