Boundary Yamabe compactness problem
Let be a smooth compact Riemannian manifold with nonempty boundary, of positive conformal type, and not conformally equivalent to the round hemisphere. Consider positive solutions of the boundary Yamabe problem, equivalently conformal metrics with constant scalar curvature and constant boundary mean curvature, normalized to remove constant rescaling. Determine, for each dimension and for general or umbilic boundary, whether the normalized solution set is compact in the appropriate Hölder topology, or whether there exist sequences of solutions developing boundary blow-up. In particular, determine the sharp dimension thresholds for the cases of scalar-flat metrics with positive constant boundary mean curvature, positive-scalar-curvature metrics with zero boundary mean curvature, and fixed real boundary mean curvature.
References
Primary source
Additional references
- On Critical Dimensions for Compactness in the Boundary Yamabe Problem, II — arXiv — Liuwei Gong, Seunghyeok Kim, Monica Musso, Juncheng Wei
Progress summary
A new preprint reports sharper dimension thresholds for compactness; the general boundary problem remains open.
The problem asks when solutions of the boundary Yamabe problem form a compact family rather than developing blow-up. Known examples show that the answer depends strongly on dimension and on curvature and boundary conditions.
Known results
- For umbilic boundaries, compactness holds for , while noncompactness examples exist for ; the result was published in 2017.
- Earlier work established compactness and blow-up results for perturbed boundary Yamabe equations, but not the full unperturbed problem.
October 7, 2026 development
Gong, Kim, Musso, and Wei report a preprint proving compactness through dimensions , , , and in specified boundary classes, together with further fixed-mean-curvature ranges. Combined with noncompactness constructions, these results claim sharper transition dimensions, but the preprint’s claims remain unverified here.
Current status (as of October 2026): Compactness through and noncompactness from are established in the earlier umbilic case, while the newer dimension ranges are unverified claims and the full problem remains open.
Sources
Solutions 0
No solutions have been posted yet.