# Some results on $\sigma_2$ curvature

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Hao Fang, University of Iowa

In this talk, we discuss some recent results on $\sigma_2$ curvature on a 4 dimensional manifold. For a conic 4-sphere, we establish some numerical necessary condition for the existence of constant $\sigma_2$ curvature conformal metric. We also discuss the boundary compactness of the solution moduli. For the hyperbolic 4-disc, we establish a Penrose type inequality, which in particular implies a “negative” mass theorem.