{
  "abstract": "Let $(M^n,g)$ be a compact Riemannian manifold of dimension $n\\ge 2$, 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 $\\partial M\\neq\\emptyset$. Given a reference scalar curvature $R_E>0$, set $K_E:=R_E/(n(n-1))$, and define the traceless Ricci tensor $\\mathring{\\Ric}:=\\Ric-\\frac{R}{n}g$ and the Einstein deviation \\[ Θ:=|\\mathring{\\Ric}|+\\frac{|R-R_E|}{n},\\qquad κ_{\\mathrm{Ein}}(p):=\\left(\\frac{1}{\\vol(M)}\\int_MΘ^p\\,dV\\right)^{1/p}, \\quad p>n/2. \\] We say that $M$ is $\\varepsilon$-nearly Einstein if $κ_{\\mathrm{Ein}}(p)<\\varepsilon$. The main result states that for every $α\\in(0,1)$ there exists $\\varepsilon_0=\\varepsilon_0(n,p,α,D,R_E)>0$ such that if $\\diam(M)\\le D$ and $κ_{\\mathrm{Ein}}(p)<\\varepsilon_0$, then \\[ λ_1(M)\\ge 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 $D\\geπ/\\sqrt{K}$. The proof combines the Bochner formula for the decomposition $\\Ric=(n-1)K_Eg+θg+\\mathring{\\Ric}$ with an auxiliary function $J$ satisfying \\[ ΔJ-τ\\frac{|\\nabla J|^2}{J}-2JΘ=-σJ \\] and Neumann boundary conditions, together with Moser iteration and a maximum principle. As $κ_{\\mathrm{Ein}}(p)\\to0$, the estimate recovers the classical Bakry--Qian theorem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04846",
  "author_total": 1,
  "authors": [
    {
      "name": "Shunzi Guo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fj6bbfpgwpvq7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04846",
  "fulltext_url": null,
  "id": "fj6bbfpgwpvq7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04846",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7220677753",
  "pdf_url": "https://arxiv.org/pdf/2610.04846",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fj6bbfpgwpvq7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.04846",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04846"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Spectral Gap Estimates on Nearly Einstein Manifolds with Integral Curvature Deviation",
  "title_text": "Spectral Gap Estimates on Nearly Einstein Manifolds with Integral Curvature Deviation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/fj6bbfpgwpvq7",
  "venue": "arXiv",
  "versions": []
}