Scalar curvature equation on SnSn, Part II: Analytic characterizations

This is the second part of a series devoting to the study of the prescribing scalar curvature problem on the standard sphere of any dimension. By studying topological degrees for certain abstract maps, we will give explicit analytic conditions on the scalar curvature function which verify the topological degree conditions given in the first part of the series to ensure the solvability of the problem. General existence results for the prescribing scalar curvature equation will be given on both H-symmetric and sub-H-symmetric solutions corresponding to H-symmetric scalar curvature functions, as well as on non-symmetric solutions corresponding to symmetric-like scalar curvature functions. Special axisymmetric and axisymmetric-like cases will be also considered. Our analysis will be based on a general approach of dimension reductions and degree calculations by taking advantage of symmetries and symmetric-like properties.