Conformal Boundary Deformations under Ricci Lower Bounds: Eigenvalue Counterexamples and Area Obstructions

Abstract: Let $(M<sup>{n+1},g)$ be a compact Riemannian manifold with boundary. A sharp strengthening of the Choi--Wang--Reilly estimate, proposed under $\Ric_g\geq ng$ and $\II_g\geq0$, would assert that the first nonzero Laplace eigenvalue of the boundary is at least $n$. We disprove this assertion in every dimension $n+1\geq3$. More precisely, we construct a sequence of metrics on the hemisphere $\Sph<sup>{n+1}_{+}$ converging in $C<sup>\infty$ to the round metric and satisfying [ \Ric_g>n g,\qquad \II_g>0,\qquad λ1(\partial\Sph^{n+1}{+},g|{\partial\Sph^{n+1}{+}})<n. ] The construction starts from Zhu's infinitesimal conformal deformation, which lowers one branch of the first boundary eigenspace while preserving the normalized Ricci lower bound to first order. We add a multiple of the spherical height function. This mode vanishes on the equator, lies in the kernel of the linearized normalized Ricci tensor, and changes the outward normal derivative by a constant. It therefore leaves the induced boundary metric and the first-order Ricci inequality unchanged while making the boundary strictly convex. A quadratic constant rescaling restores the exact Ricci bound without destroying the linear eigenvalue decrease. We also study the boundary-area conjecture under $\Ric\geq0$ and $\II\geq g|_{\partial M}$. For conformal deformations of the Euclidean ball satisfying the corresponding first-order Ricci and convexity inequalities, we obtain a quantitative nonpositive upper bound for the right first variation of Hui-Hsien Wang's scale-invariant boundary-convexity functional. Equality occurs exactly for affine infinitesimal conformal factors. In particular, the recent conformal counterexamples to Escobar's Steklov conjecture move strictly away, to first order, from violating the boundary-area conjecture.