Compactness of singular solutions to the GJMS equation

Início: 22/03/2024 17:00

Término: 22/03/2024 18:00

Palestrante: João Henrique Santos de Andrade (USP)

E-mail do Palestrante: andradejh@ime.usp.br

Resumo: We study some compactness properties of the set of conformally flat singular metrics with constant positive $Q$-curvature (integer or fractional) on a finitely punctured sphere. Based on some recent classification results, we focus on some cases of integer $Q$-curvature. We introduce a notion of necksize for these metrics in our moduli space, which we use to characterize compactness. More precisely, we prove that if the punctures remain separated and the necksize at each puncture is bounded away from zero along a sequence of metrics, then a subsequence converges with respect to the Gromov-Hausdorff metric. Our proof relies on an upper bound estimate which is proved using moving planes and a blow-up argument. This is combined with a lower bound estimate which is a consequence of a removable singularity theorem. We also introduce a homological invariant which may be of independent interest for upcoming research.