Spectral Gap Estimates on Nearly Einstein Manifolds with Integral Curvature Deviation
Abstract
Let (Mn,g) be a compact Riemannian manifold of dimension n≥ 2, possibly with nonempty convex C2 boundary. We study the first nonzero eigenvalue λ1(M) of the Laplace--Beltrami operator, subject to the Neumann boundary condition when ∂ M≠∅. Given a reference scalar curvature RE>0, set KE:=RE/(n(n−1)), and define the traceless Ricci tensor mathringRic:=Ric−R/ng and the Einstein deviation Θ:=|mathringRic|+(|R−RE|)/n, κEin(p):=(1/vol(M)∫MΘp dV)1/p, p>n/2. We say that M is ε-nearly Einstein if κEin(p)<ε. The main result states that for every α∈(0,1) there exists ε0=ε0(n,p,α,D,RE)>0 such that if diam(M)≤ D and κEin(p)<ε0, then λ1(M)≥ α λ1(n,KE,D), where λ1(n,K,D) is the first nonzero eigenvalue of the Bakry--Qian one-dimensional model problem with curvature parameter K, with the convention λ1(n,K,D)=nK when K>0 and D≥π/√K. The proof combines the Bochner formula for the decomposition Ric=(n−1)KEg+θg+mathringRic with an auxiliary function J satisfying ΔJ−τ(|∇ J|2)/J−2JΘ=−σJ and Neumann boundary conditions, together with Moser iteration and a maximum principle. As κEin(p)→0, the estimate recovers the classical Bakry--Qian theorem.
Related
- Einstein four-manifolds of positive sectional curvature and ADM mass
- K{ä}hler Einstein manifolds and the Calabi curvature operator
- Stability of Einstein 4-manifolds satisfying a chiral curvature condition
- Inhomogeneous Strichartz estimates on manifolds with nonpositive curvature and applications
- Convergence of cohomogeneity-one Lagrangian mean curvature flow in positive Kähler-Einstein manifolds
BibTeX
@misc{guo2026spectral,
title = {{Spectral Gap Estimates on Nearly Einstein Manifolds with Integral Curvature Deviation}},
author = {Shunzi Guo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04846},
eprint = {2610.04846},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04846},
}
RIS
TY - UNPB
TI - Spectral Gap Estimates on Nearly Einstein Manifolds with Integral Curvature Deviation
AU - Shunzi Guo
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.04846
UR - https://arxiv.org/abs/2610.04846
AB - Let (M^n,g) be a compact Riemannian manifold of dimension n≥ 2, possibly with nonempty convex C^2 boundary. We study the first nonzero eigenvalue λ_1(M) of the Laplace--Beltrami operator, subject to the Neumann boundary condition when ∂ M≠∅. Given a reference scalar curvature R_E>0, set K_E:=R_E/(n(n−1)), and define the traceless Ricci tensor mathringRic:=Ric−R/ng and the Einstein deviation Θ:=|mathringRic|+(|R−R_E|)/n, κ_Ein(p):=(1/vol(M)∫_MΘ^p dV)^1/p, p>n/2. We say that M is ε-nearly Einstein if κ_Ein(p)<ε. The main result states that for every α∈(0,1) there exists ε_0=ε_0(n,p,α,D,R_E)>0 such that if diam(M)≤ D and κ_Ein(p)<ε_0, then λ_1(M)≥ α λ_1(n,K_E,D), where λ_1(n,K,D) is the first nonzero eigenvalue of the Bakry--Qian one-dimensional model problem with curvature parameter K, with the convention λ_1(n,K,D)=nK when K>0 and D≥π/√K. The proof combines the Bochner formula for the decomposition Ric=(n−1)K_Eg+θg+mathringRic with an auxiliary function J satisfying ΔJ−τ(|∇ J|^2)/J−2JΘ=−σJ and Neumann boundary conditions, together with Moser iteration and a maximum principle. As κ_Ein(p)→0, the estimate recovers the classical Bakry--Qian theorem.
ER -