\(\sigma_2\) Yamabe problem on conic 4-spheres (Q2423885)

On a given Riemannian manifold, the Yamabe problem consists in finding, in the same conformal class, a metric whose scalar curvature is constant. The authors consider here aspects of the analoguous problem for the \(\sigma_2\) invariant, built as the second symmetric sum of the eigenvalues of the Schouten tensor. More precisely, they study this problem in the case of 4-spheres equipped with conformally flat metrics with conic singularities. Their main result gives a necessary condition (no supercriticality) for the existence of a solution to the \(\sigma_2\)-Yamabe problem. The proof relies in part on the Gauss-Bonnet-Chern integral formula established by the authors for a class of metrics on conic spheres. For Part II, see [the authors, Pac. J. Math. 311, No. 1, 33--51 (2021; Zbl 1492.53049)].