Choi–Wang–Reilly boundary eigenvalue conjecture and Hui-Hsien Wang boundary-area conjecture

The problem concerns a proposed sharp boundary eigenvalue bound and a related boundary-area inequality for geometric spaces with boundary. No proposer or date is identified in the retrieved material.

Known results

• A December 2015 paper constructed, for every $n\geq 2$, positively Ricci-curved hemispheres with minimal boundary and first boundary eigenvalue $\lambda_1<n$, giving a boundary counterexample to the relevant Choi–Wang conjecture.
• The same work established a first-variation rigidity statement: under technical hypotheses fixing Ricci curvature and boundary mean curvature to first order, the first boundary eigenvalue has zero first variation.
• A later result extended a theorem of Choi and Wang to nonminimal hypersurfaces, without resolving either conjecture.

August 26, 2026 claimed counterexample and area obstruction

A paper titled Conformal Boundary Deformations under Ricci Lower Bounds: Eigenvalue Counterexamples and Area Obstructions claims that the proposed eigenvalue bound fails in every dimension at least three, even for metrics arbitrarily close to the round hemisphere. It also claims a first-variation obstruction to a conformal boundary-area conjecture, not a full counterexample; both claims remain unverified.

Current status (as of August 2026): The eigenvalue strengthening is claimed false in dimensions $n\geq 3$, while the boundary-area conjecture remains unresolved beyond an unverified infinitesimal obstruction.