Determine whether higher-order deformations can produce a boundary-area counterexample along affine conformal equality directions

Determine whether a second-order deformation of the Euclidean ball, tangent at first order to the affine conformal equality directions of the normalized boundary-convexity functional, can violate the boundary-area inequality under nonnegative Ricci curvature and uniformly convex boundary.

Background

Theorem 1.3 shows that every non-affine admissible first-order conformal direction strictly decreases the scale-invariant functional Λ, while affine infinitesimal conformal factors are exactly the equality directions. Consequently, the paper’s first-order analysis rules out transverse conformal instability but does not analyze higher-order behavior along the affine kernel.

Remark 5.5 identifies a specific unresolved possibility: a second-order deformation tangent to those affine equality directions might behave differently and could potentially contribute to a counterexample to the boundary-area conjecture.

References

Theorem 1.3 leaves open the possibility of a second-order deformation tangent to the affine equality directions, as well as genuinely nonconformal or global constructions. Thus a counterexample to the boundary-area conjecture, if one exists, must evade the transverse first-order conformal mechanism considered here.

— Conformal Boundary Deformations under Ricci Lower Bounds: Eigenvalue Counterexamples and Area Obstructions  (2608.25391 - Li et al., 26 Aug 2026) in Remark 5.5, Section 5, page 10