Conformal Boundary Deformations under Ricci Lower Bounds: Eigenvalue Counterexamples
Abstract
Let (Mn+1,g) be a compact Riemannian manifold with boundary. Under the assumptions Ricg≥ ng and IIg≥0, Wang proposed a sharp strengthening of the Choi--Wang--Reilly estimate, asserting that the first nonzero Laplace eigenvalue of the boundary is at least n; see [J. Geom. Anal. 31 (2021)]. We disprove this assertion in every dimension n+1≥3. More precisely, we construct a sequence of metrics on the hemisphere Sphn+1+ converging in C∞ to the round metric and satisfying Ricg>n g, IIg>0, λ1(∂Sphn+1+,g|∂Sphn+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.
Related
- Robin eigenvalue ratios: monotonicity on boxes and counterexamples
- Counterexamples to a higher-index Dirichlet eigenvalue-ratio conjecture
- Markov maps and quasi-conformal deformations associated to geometrically finite Kleinian groups
- Metric Correspondences and Low-Rank Deformations for Conformal and m-Modified Conformal Vector Fields
- Planar Dirichlet eigenvalue bounds from a trace identity
BibTeX
@misc{li2026conformal,
title = {{Conformal Boundary Deformations under Ricci Lower Bounds: Eigenvalue Counterexamples}},
author = {Fagui Li and Yuhang Zhao},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.25391},
eprint = {2608.25391},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.25391},
}
RIS
TY - UNPB
TI - Conformal Boundary Deformations under Ricci Lower Bounds: Eigenvalue Counterexamples
AU - Fagui Li
AU - Yuhang Zhao
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.25391
UR - https://arxiv.org/abs/2608.25391
AB - Let (M^n+1,g) be a compact Riemannian manifold with boundary. Under the assumptions Ric_g≥ ng and II_g≥0, Wang proposed a sharp strengthening of the Choi--Wang--Reilly estimate, asserting that the first nonzero Laplace eigenvalue of the boundary is at least n; see [J. Geom. Anal. 31 (2021)]. We disprove this assertion in every dimension n+1≥3. More precisely, we construct a sequence of metrics on the hemisphere Sph^n+1_+ converging in C^∞ to the round metric and satisfying Ric_g>n g, II_g>0, λ_1(∂Sph^n+1_+,g|_∂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.
ER -