An explicit mathematical deformation of the unit three-ball gives a counterexample to Escobar's proposed lower bound for the first nonzero Steklov eigenvalue, a number used to describe how geometry interacts with its boundary. In the construction, that eigenvalue is smaller than c(epsilon), the smallest principal curvature of the boundary, even though the metric has positive Ricci curvature and the boundary is strictly convex for every sufficiently small positive epsilon.
The manuscript identifies the example as a counterexample to Escobar's conjecture in dimension three. It is an arXiv version 1 manuscript dated 25 August 2026.
The geometry behind the test
The comparison starts with Euclidean geometry. For the unit three-ball, the boundary unit sphere has shape operator S = I, where the shape operator is the object used here to record boundary curvature. Both the first nonzero Steklov eigenvalue and the minimum eigenvalue of that boundary operator are 1. That shared value is the reference point for the perturbation.
To test the proposed inequality, the authors add epsilon times an explicit tensor to the Euclidean metric. The resulting family is an additive, nonconformal polynomial perturbation, and the tensor is a Cartesian polynomial of degree six. Epsilon is taken sufficiently small and positive.
The Ricci-curvature check is symbolic. It differentiates the linearized Ricci tensor and uses Sylvester's criterion, a test for positive definiteness, to reduce the problem to inequalities involving P, Rss and the determinant Delta, together with polynomial-positivity arguments. On that basis, the perturbed metric has positive Ricci curvature for sufficiently small positive epsilon.
At the boundary, the minimum principal curvature is reported as c(epsilon) = 1 + eta epsilon + O(epsilon squared), which is greater than 1 for sufficiently small positive epsilon. The coefficient eta is positive but is not given numerically, and the finite-epsilon remainder is not quantified. The result places the comparison threshold above its Euclidean reference value while keeping the boundary strictly convex.
A quotient that stays at one
The decisive estimate uses the coordinate function s as a trial function in a Rayleigh quotient, the variational quantity used to estimate a Steklov eigenvalue. Its first-order variation is zero, so the perturbed quotient is 1 + O(epsilon squared). The calculation then gives sigma1 for the perturbed three-ball as smaller than c(epsilon) for every sufficiently small positive epsilon.
This is the point at which the conjectured lower bound fails: the boundary's minimum curvature sits above the ball's first nonzero Steklov eigenvalue. The result is stated only in a small-positive-epsilon regime. The paper gives no explicit numerical upper threshold for epsilon, and its second-order expansions do not come with quantified finite-epsilon error bounds.
A second result on a disk
A separate calculation examines a specified disk in the construction. For every sufficiently small positive epsilon, Sigma is a properly embedded, totally geodesic free-boundary minimal disk, and its first nonzero Steklov eigenvalue obeys sigma1(Sigma) <= 1 - (93739/35481600)epsilon + O(epsilon squared) < 1.
The disk estimate is also limited to sufficiently small positive epsilon, and no explicit interval for epsilon is reported. Its second-order remainder is likewise not quantified, so the displayed expansion is not a full finite-parameter error bound.
What the construction leaves open
The construction also has a clear limit. The manuscript states that the example leaves the class Sec >= 0, meaning metrics with nonnegative sectional curvature. The counterexample therefore does not establish failure of the bound under that additional sectional-curvature condition.
The evidence is confined to one explicitly defined family on the unit three-ball. It does not establish the same behavior for arbitrary manifolds, and the admissible range of epsilon is not fully quantified.
The perturbation tensor is given explicitly in Section 3 through equations (6) to (11), and no separate dataset is described.
The manuscript reports no funding source. It also discloses extensive ChatGPT use for mathematical discussions and editorial assistance.
Paper data and sources
Original title: An elementary counterexample to Escobar's Steklov conjecture on the three-ball
Authors: Alexandre Girouard, Thomas Hélière
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text