Interior C² estimate problem for the graphical scalar-curvature equation

For every dimension $n\ge 3$ , let $\Omega\subset\mathbb{R}^{n}$ be a domain and let $u\in C^{3}(\Omega)$ be admissible, meaning that the principal-curvature vector of its graph $M_{u}=\{(x,u(x)):x\in\Omega\}\subset\mathbb{R}^{n+1}$ lies in the Gårding cone $\Gamma_{2}$ . If $M_{u}$ has constant scalar curvature, equivalently $\operatorname{Scal}_{M_{u}}=c$ for a constant $c$ , establish interior curvature estimates: on every $\Omega'\Subset\Omega$ , bound the norm of the second fundamental form $|A|$ on $M_{u}|_{\Omega'}$ in terms of the prescribed data, with no restriction on the dimension other than $n\ge 3$ . Such estimates should yield the corresponding interior $C^{2}$ bounds and regularity for admissible graphical solutions.

References

Primary source

arXiv

Additional references

Interior Curvature Estimates for the Graphical Scalar Curvature Equation in All Dimensions — arXiv

elliptic-equations geometric-analysis partial-differential-equations

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A new unrefereed preprint claims to settle the interior regularity problem in every dimension, but the result has not been independently verified.

The problem asks for dimension-free interior curvature bounds and regularity for graphical solutions of the scalar-curvature equation. Earlier literature treated only special dimensions or solution classes; a new preprint now claims the unrestricted all-dimensional result.

Known results

In dimension $n=3$ , the Hessian estimate is due to Warren and Yuan, with extensions by Qiu, Zhou, and Xu; Qiu obtained the corresponding curvature estimate.
For $n=4$ , Shankar and Yuan established the Hessian-equation estimate.
Guan and Qiu (2017) proved interior estimates under convexity or weakened structural hypotheses, not for the unrestricted problem.
A 2026 preprint reported only a partial dimension- $4$ curvature result and explicitly described the general problem as open.

September 2, 2026 claimed all-dimensional estimate

An arXiv preprint claims all-dimensional interior curvature estimates, using Jacobi inequalities, a two-surface maximum principle, and a two-surface Pogorelov estimate. If correct, this resolves the long-standing regularity problem without a dimensional restriction; the claim is currently supported only by an unrefereed preprint.

Current status (as of September 2026): Special cases in dimensions $n=3$ and $n=4$ are known, while the advertised all-dimensional resolution remains an unverified claim.

Sources

No solutions have been posted yet.