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Conformal Ricci Curvature and Spectral Estimates
Abstract
We derive optimized upper bounds for the first Dirichlet eigenvalue of a bounded domain under lower bounds for the conformal Ricci tensor of Shen and Ye. The method also yields sharp estimates for the bottom spectrum on complete noncompact manifolds, a four-dimensional application involving Q-curvature, and local and global spectral-Ricci extensions of Cheng's estimate. As a consequence, we obtain a sharp comparison between the spectra of the Laplacian and the conformal Laplacian in terms of the smallest Schouten eigenvalue.
Related
- Jacobian estimates and geometric inequalities under intermediate Ricci curvature
- Gradient estimates for the Green kernel under spectral Ricci bounds, and the stable Bernstein theorem in ℝ⁴
- Spectral Gap Estimates on Nearly Einstein Manifolds with Integral Curvature Deviation
- Topological regularity and stability of noncollapsed spaces with Ricci curvature bounded below
- Kähler-Ricci Shrinkers with Nonnegative Ricci Curvature
BibTeX
@misc{jin2026conformal,
title = {{Conformal Ricci Curvature and Spectral Estimates}},
author = {Xiaoshang Jin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02104},
eprint = {2609.02104},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02104},
}
RIS
TY - UNPB
TI - Conformal Ricci Curvature and Spectral Estimates
AU - Xiaoshang Jin
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02104
UR - https://arxiv.org/abs/2609.02104
AB - We derive optimized upper bounds for the first Dirichlet eigenvalue of a bounded domain under lower bounds for the conformal Ricci tensor of Shen and Ye. The method also yields sharp estimates for the bottom spectrum on complete noncompact manifolds, a four-dimensional application involving Q-curvature, and local and global spectral-Ricci extensions of Cheng's estimate. As a consequence, we obtain a sharp comparison between the spectra of the Laplacian and the conformal Laplacian in terms of the smallest Schouten eigenvalue.
ER -