Boundedness of the three-dimensional weak-cone Yamabe constants

Determine whether the quantities Y_{2,1}(M,[g]) and Y_{2,0}(M,[g]) are bounded from above for arbitrary three-dimensional locally conformally flat manifolds, and determine whether the analogous quantities \bar Y_{2,1}(M,[g]) and \bar Y_{2,0}(M,[g]) are likewise bounded from above.

Background

The paper introduces Yamabe-type constants defined using the cones \mathcal C_{k-1} and \mathcal C_k. In dimension three with k=2, upper boundedness of these constants is relevant to the existence and analysis of minimizers for the quotient involving total c3-curvature.

The authors prove affirmative boundedness results for locally conformally flat manifolds through their flow method, but explicitly state that the general boundedness questions remain unresolved. The analogous questions for the constants defined on the narrower cone are also left open.

References

It remains open in general whether, for $n=3$, the quantities $Y_{2,1}(M,[g])$ and $Y_{2,0}(M,[g])$ are bounded from above. The analogous questions for $\bar Y_{2,1}$ and $\bar Y_{2,0}$ are also open.

— Optimal geometric inequalities and fully nonlinear conformal flows  (2609.11421 - Ge et al., 10 Sep 2026) in Section 1, paragraph immediately following the discussion of Theorem 1.6