Scalar curvature equation on SnSn, Part I: Topological conditions

This is the first part of a series devoting to the study of the prescribing scalar curvature problem on the standard sphere of any dimension. In the first part, we will adopt the degree-theoretic approach to give a topological condition and some general, explicit conditions on the scalar curvature functions to ensure the solvability of the problem. Our topological condition is imposed on some of simple maps explicitly defined by the scalar curvature function, which is derived from the asymptotic expansion of the boundary map introduced in [A. Chang, P. Yang, A perturbation result in prescribing scalar curvature on SnSn, Duke Math. J. 64 (1991) 27–69]. Our conditions, particularly allowing non-isolation and non-degeneracy of the critical points of the scalar curvature functions, can be easily verified in many situations. In the second part of series, we will make a detailed study on the verification of the topological condition. Our results will generalize almost all existing ones in the same direction and meanwhile provide a unified treatment for both symmetric and non-symmetric cases of the scalar curvature functions.