Yamabe–Escobar compactness conjecture

Let $n\geq 3$ and let $(M^n,g)$ be a compact Riemannian manifold with nonempty boundary and positive Yamabe–Escobar invariant. Consider the normalized set of conformal metrics $\mathcal{S}(M,[g])=\{\widetilde g=u^{4/(n-2)}g:u\in C^\infty(M),\ u>0,\ \operatorname{Vol}_{\widetilde g}(M)=1,\ R_{\widetilde g}\equiv c_{\widetilde g}>0,\ H_{\widetilde g}\equiv 0\text{ on }\partial M\}$ . The compactness conjecture asserts that if $(M,[g])$ is not conformally equivalent to the round hemisphere $(\mathbb{S}^n_+, [g_{\mathrm{round}}])$ , then $\mathcal{S}(M,[g])$ is compact in the appropriate normalized solution topology.

References

Primary source

arXiv

Additional references

Compactness and Noncompactness for the Positive Yamabe--Escobar Problem with Minimal Boundary — arXiv — Santiago Cordero-Misteli

Posted 1 day ago · edited 1 day ago

Progress summary

Refreshed 1 day ago

Claimed progress

A new unrefereed paper claims sharp compactness boundaries, but the general conjecture remains unverified.

The conjecture concerns compactness of solution sets for the Yamabe problem on manifolds with boundary, together with related Weyl-tensor vanishing conditions. The latest paper claims compactness for $3\le n\le14$ and noncompactness beginning at dimensions $15$ , $19$ , and $21$ in different settings.

Known results

Umbilic-boundary compactness was proved for $3\le n\le24$ , with counterexamples for $n\ge25$ (2012).
Conditional compactness results use nonvanishing Weyl curvature on the boundary, including a separate $n=8$ condition.
Earlier constructions give noncompactness for nonumbilic boundaries in dimensions $N\ge15$ and for umbilic boundaries in dimensions $N\ge21$ in a related positive-curvature setting.

Recent threshold claim (2026)

Santiago Cordero-Misteli’s preprint claims the stated compactness range and sharp thresholds, including associated Weyl–umbilicity vanishing results. The retrieved evidence provides no independent mathematical assessment, so this remains an unverified claim rather than an established resolution.

Current status (as of October 2026): The claimed thresholds $15$ , $19$ , and $21$ are not independently verified; the general Yamabe–Escobar compactness conjecture remains open in the evidence retrieved.

Sources

No solutions have been posted yet.