Application of the method of moving planes to conformally invariant equations
A basic question in Riemannian geometry is to determine to what extent curvature assumptions on the Riemannian metric \(g\) impose restrictions on the topology of \((M^n,g)\). Such a question is mainly motivated by the Gauss-Bonnet theorem for two-dimensional compact manifolds: \[ 2\pi\chi(M^2)=\int_{M^2}K\,d\mu\tag{1} \] where \(d\mu\) is the area element, \(K\) is the Gauss curvature, and \(\chi(M^2)\) is the Euler characteristic. Motivated by such a question A. Chang, M. Gursky and P. Yang asked the following: Given a compact Riemannian 4-manifold \((M^4,g_0)\) with both \(\int_{M^4} \sigma_2(A_0)\,dv_0\) and Yamabe invariant \(Y(g_0)=\inf_{g\in[g_0]} (\text{vol}(g))^{-1/2} \int_{M^4} R(g)\,dv_g\) positive, where \([g_0]\) denotes the conformal class of \(g_0\). Does there exist a conformal metric \(g=e^{2w}g_0\) with positive Ricci curvature Ric\(_g\)? Here \(A =\text{Ric}-{1\over 6}Rg\) denotes the Schouten tensor, \(\sigma_2(A)=-{1\over 2}| \text{Ric}| ^2+{1\over 6}R^2\) and \(R\) is the scalar curvature. In [\textit{S.-Y. A. Chang, M. J. Gursky}, and \textit{P. C. Yang}, Ann. Math. (2) 155, No. 3, 709--787 (2002; Zbl 1031.53062)], they gave a positive answer to the above question under the stronger assumption that there is a conformal metric \(g\) with \(\sigma_2(A)>0\). The authors of the paper under review prove a high dimensional version of the above result under the assumption that the manifold is locally conformally flat. Their proof uses the method of moving planes and the fundamental result of \textit{R. Schoen} and \textit{S.-T. Yau} [Invent. Math. 92, No. 1, 47--71 (1988; Zbl 0658.53038)] on developing maps for locally conformally manifolds. They also give a generalization of the above result for more general conformally invariant equations.
scientific article; zbMATH DE number 2133469
scientific article