Remove conformal flatness from the three-dimensional c3-curvature inequalities

Determine whether the inequalities in Corollary 1.4 for three-dimensional locally conformally flat manifolds remain valid without the conformal flatness assumption.

Background

For k=2 and n=3, the paper establishes two sharp inequalities involving the total c3-curvature on locally conformally flat 3-manifolds. These inequalities answer Conjectures 3 and 4 of the cited work in that setting and are related to a question of Viaclovsky.

The unresolved issue is whether local conformal flatness is genuinely necessary or whether the same inequalities hold on arbitrary three-dimensional manifolds in the relevant conformal class.