A mathematical preprint reports a counterexample to a proposed boundary rule. In every ambient dimension n + 1 that is at least 3, its constructed metrics converge smoothly to the round hemisphere, satisfy strict Ricci curvature and strict boundary convexity, and still have a first nonzero boundary Laplace eigenvalue below n. The eigenvalue is the first nonzero spectral value associated with the boundary, so the result directly challenges the proposed threshold.
The rule under examination asks whether a Ricci lower bound of n, a minimum condition on the space's internal bending, together with a nonnegative boundary second fundamental form, a condition on how the edge bends, implies that the first nonzero boundary eigenvalue is at least n. The construction analyzes a sequence of smooth metrics on the hemisphere for every integer n at least 2, with the sequence converging to the round metric. Each constructed boundary is also umbilic, meaning its curvature has the same value in every tangent direction.
A deformation tuned to the boundary
To build the sequence, the analysis uses a conformal deformation, meaning a controlled change to the metric through a smooth factor. The Zhu deformation supplies a smooth conformal factor with nonnegative linearized normalized Ricci tensor and an analytic boundary-eigenvalue branch with a negative derivative. In ordinary terms, the selected spectral branch initially moves downward while the first-order Ricci condition remains nonnegative.
The construction then adds a sufficiently large multiple of the spherical height mode. This boundary-invisible mode leaves the induced boundary metric and the first-order Ricci condition unchanged while making the boundary strictly convex. Combined with the eigenvalue deformation, it produces the strict curvature inequalities and below-n eigenvalue reported for the small members of the sequence.
One final adjustment matters for the curvature claim. A constant rescaling of quadratic order turns the first-order Ricci inequality into a strict Ricci lower bound for sufficiently small positive t. The eigenvalue construction is likewise perturbative, with the below-n conclusion applying to sufficiently small members of the sequence.
The area conjecture faces narrower tests
The preprint also examines a separate conjecture about boundary area. It asks whether nonnegative Ricci curvature and boundary principal curvatures of at least 1 imply an area no larger than that of the unit n-sphere. To study the question locally, the analysis follows a twice-differentiable curve of smooth conformal metrics on the Euclidean ball and its initial conformal direction.
For the admissible directions satisfying Aφ ≥ 0 and qφ ≥ 0, the right first derivative of the normalized boundary-area functional Λ is at most zero. The paper gives this as a negative normalized trace-integral bound, but the message is simple: along these first-order conformal directions, Λ does not increase. That rules out a first-order conformal rise in the tested setting, not the full area conjecture.
The equality case is narrow. The right derivative vanishes exactly for affine infinitesimal conformal factors. Under qφ ≥ 0, the constant term is nonpositive, and a nonzero Aφ gives a strict decrease. Outside those affine equality directions, the first-order area functional therefore moves downward.
Other constructions stay below the area threshold
The conformal calculation is also applied to Steklov examples, another boundary spectral construction. Conditional on the stated full perturbative criterion, Λ is below 1 for sufficiently small positive t. Both the unrescaled metrics and the versions normalized by the least boundary curvature have boundary area below the unit-sphere area.
A separate check looks at smooth rotationally symmetric ball metrics. When such a metric has nonnegative Ricci curvature and boundary principal curvature at least 1, its boundary area is no greater than the unit-sphere area, with equality only for the Euclidean unit ball. Because this result is confined to the rotationally symmetric class, it does not settle the general area question.
The broader questions remain open
Taken together, the findings leave the sharp eigenvalue constant and the broader area problem unresolved. The optimal universal eigenvalue constant remains undetermined, with a lower bound of n/2 and an upper bound below n. On area, the analysis rules out an increase along the tested first-order conformal directions and proves the unit-sphere bound in the rotationally symmetric class, but it does not establish the general boundary-area conjecture. Second-order behavior along the affine equality directions and more general nonconformal or anisotropic constructions remain outside these results.
The work is an arXiv version 1 preprint dated 26 Aug 2026. It analyzes constructed metric families on a hemisphere, a Euclidean ball and rotationally symmetric ball metrics rather than a human or observational dataset. Its end matter lists affiliations and email addresses but reports no funding statement.
Paper data and sources
Original title: Conformal Boundary Deformations under Ricci Lower Bounds: Eigenvalue Counterexamples and Area Obstructions
Authors: Fagui Li, Yuhang Zhao
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text