Rigidity of Einstein Manifolds under Two Eigenvalue Conditions on the Curvature Operator of the Second Kind
Abstract
We prove a rigidity theorem for closed Einstein manifolds of dimension n≥6 under two lower bounds on the curvature operator of the second kind. More precisely, for L1, L2≥0 and some real number α>1 in a suitable range, we consider λ1≥−L1λ̅, 1/α∑j=1αλj ≥−L2λ̅. Here λ1≤⋯≤λN are the eigenvalues of the curvature operator of the second kind mathring R, N=(n−1)(n+2)/2, and λ̅=N−1∑j=1Nλj. Let $θ(n, α)$ be the constant appearing in \cite[Theorem~1.1]{CW26}; see (2). In the parameter range considered here, L2>θ(n, α), so the second condition is strictly weaker than the corresponding condition in \cite{CW26}, while the first condition and that condition do not imply each other. Under an additional explicit relation between L1 and L2, we prove that the manifold is either flat or a spherical space form.
Related
- Eigenvalue comparison theorems for the Witten-Laplacian and the weighted p-Laplacian on complete manifolds with a modified Ricci curvature bounded from below
- Dirichlet eigenvalue asymptotics on conformally compact Einstein manifolds
- Scalar and tensor eigenvalue bounds for conformally Kähler Einstein four-manifolds
- Einstein four-manifolds of positive sectional curvature and ADM mass
- K{ä}hler Einstein manifolds and the Calabi curvature operator
BibTeX
@misc{cheng2026rigidity,
title = {{Rigidity of Einstein Manifolds under Two Eigenvalue Conditions on the Curvature Operator of the Second Kind}},
author = {Haiqing Cheng and Kui Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23597},
eprint = {2609.23597},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23597},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rigidity of Einstein Manifolds under Two Eigenvalue Conditions on the Curvature Operator of the Second Kind
AU - Haiqing Cheng
AU - Kui Wang
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23597
UR - https://arxiv.org/abs/2609.23597
AB - We prove a rigidity theorem for closed Einstein manifolds of dimension n≥6 under two lower bounds on the curvature operator of the second kind. More precisely, for L_1, L_2≥0 and some real number α>1 in a suitable range, we consider λ_1≥−L_1λ̅, 1/α∑_j=1^αλ_j ≥−L_2λ̅. Here λ_1≤⋯≤λ_N are the eigenvalues of the curvature operator of the second kind mathring R, N=(n−1)(n+2)/2, and λ̅=N^−1∑_j=1^Nλ_j. Let $θ(n, α)$ be the constant appearing in \cite[Theorem~1.1]{CW26}; see (2). In the parameter range considered here, L_2>θ(n, α), so the second condition is strictly weaker than the corresponding condition in \cite{CW26}, while the first condition and that condition do not imply each other. Under an additional explicit relation between L_1 and L_2, we prove that the manifold is either flat or a spherical space form.
ER -