Log-removal problem for endpoint eigenfunction restriction estimates

Let $(M,g)$ be a smooth closed Riemannian manifold of dimension at least $3$ , let $\Sigma\subset M$ be a fixed smooth submanifold of codimension $2$ , and let $\phi_\lambda$ be an $L^2(M)$ -normalized Laplace eigenfunction satisfying $-\Delta_g\phi_\lambda=\lambda^2\phi_\lambda$ . Does there exist a constant $C_{M,\Sigma}$ such that, for all $\lambda\geq 1$ , $\|\phi_\lambda|_\Sigma\|_{L^2(\Sigma)}\leq C_{M,\Sigma}\lambda^{1/2}$ ? Equivalently, can the logarithmic factor in the general endpoint estimate $\|\phi_\lambda|_\Sigma\|_{L^2(\Sigma)}=O\!\left(\lambda^{1/2}\sqrt{\log\lambda}\right)$ be removed for all such $(M,\Sigma)$ ?

References

Primary source

arXiv

Additional references

Endpoint eigenfunction restriction estimates in codimension two — arXiv — Xing Wang, Cheng Zhang

harmonic-analysis partial-differential-equations spectral-geometry

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed solved

A new paper claims the logarithmic loss cannot be removed in general, while improving the universal bound, but the claim has not been independently verified.

The problem asks whether the endpoint logarithmic loss in restriction estimates for codimension-two submanifolds can be removed in the general Riemannian setting. Earlier work established the issue in special geometries; the September 2026 paper by Xing Wang and Cheng Zhang claims a negative answer in general.

Known results

Burq–Gérard–Tzvetkov and Hu obtained sharp estimates apart from the endpoint logarithmic loss.
Chen–Sogge removed the loss for geodesics on three-dimensional manifolds.
Wang–Zhang (2020) removed it for totally geodesic codimension-two submanifolds in arbitrary dimensions, and for several curved-curve cases in dimension three.

September 2026 claimed negative resolution

Wang and Zhang claim $o(\lambda^{1/2}\sqrt{\log \lambda})$ for every fixed codimension-two submanifold and construct examples ruling out the log-free $O(\lambda^{1/2})$ estimate in the general setting. This is a claimed resolution, not an independently verified one.

Current status (as of September 2026): The general problem is claimed to be settled negatively, with a sharper universal upper estimate, but that claim remains unverified; special geometries may still admit log-free bounds.

Sources

No solutions have been posted yet.