{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "35FFCR3GRH"
    },
    {
      "field": "uses",
      "target": "F4WDYPQ9VP"
    },
    {
      "field": "uses",
      "target": "SSCJWAQ7AX"
    }
  ],
  "body": "\n## Statement\n\nLet $(M, g)$ be a closed [[Riemannian manifold|7CBT4SM83Y]] of dimension $n$, and let $K(t, x, y)$ be its [[heat kernel|327NP6S7H5]].\nThen there are smooth functions $a_0, a_1, a_2, \\ldots$ on $M$ such that for every integer $N \\ge 0$\n$$K(t, x, x) = (4\\pi t)^{-n/2} \\left( \\sum_{k = 0}^{N} a_k(x) \\, t^k + O(t^{N+1}) \\right) \\quad \\text{as } t \\to 0^+,$$\nuniformly in $x \\in M$.\nEach $a_k(x)$ is a universal polynomial in the curvature tensor and its covariant derivatives at $x$.\nThe first two are $a_0 = 1$ and $a_1 = \\tfrac16 S$, where $S$ is the [[scalar curvature|ZHX2VYQDZ3]].\nIntegrating over $M$, the heat trace satisfies\n$$\\sum_{j \\ge 0} e^{-t\\lambda_j} \\sim (4\\pi t)^{-n/2} \\sum_{k \\ge 0} t^k \\int_M a_k \\, dv,$$\nwhere $\\lambda_0 \\le \\lambda_1 \\le \\cdots$ is the [[spectrum of the Laplace–Beltrami operator|B8TYKV7AA1]].\n\n## Remarks\n\nIdea of the proof: build an approximate heat kernel near the diagonal as the Euclidean Gaussian in normal coordinates times a power series $\\sum u_k(x, y) t^k$.\nThe coefficients $u_k$ solve transport equations along geodesics, and $a_k(x) = u_k(x, x)$.\nA Duhamel argument shows that the true heat kernel differs from the truncated approximation by $O(t^{N + 1 - n/2})$.\nConsequences: the spectrum determines the dimension $n$, the volume $\\int_M a_0 = \\operatorname{vol}(M)$ and the total scalar curvature $\\int_M S$.\nThe next integrated coefficient is $\\int_M a_2 = \\frac{1}{360} \\int_M \\left( 2\\abs{\\mathrm{Rm}}^2 - 2\\abs{\\mathrm{Ric}}^2 + 5 S^2 \\right) dv$.\nExample: on a flat torus $\\R^n / \\Gamma$, $K(t, x, x) = (4\\pi t)^{-n/2} \\sum_{\\gamma \\in \\Gamma} e^{-\\abs{\\gamma}^2 / 4t}$, so $a_0 = 1$ and $a_k = 0$ for $k \\ge 1$, with an exponentially small error.\n\n## Explanations\n\n### L1 human\n\nFor a very short time, heat spreading from a point only feels a tiny neighbourhood, which looks almost flat.\nSo the heat kernel at the point is close to the flat-space value $(4\\pi t)^{-n/2}$.\nThe corrections come in powers of $t$, and each coefficient is computed from the curvature at the point.\n\n### L2 human\n\nMinakshisundaram and Pleijel used the expansion to continue the spectral zeta function of the Laplacian meromorphically.\nIt shows that the spectrum determines the volume and the total scalar curvature.\nThis answers some spectral rigidity questions.\nFor example, a closed surface with the spectrum of the round sphere is a round sphere.\nThe same construction works for Laplace-type operators on vector bundles, and the coefficients there are the Seeley–DeWitt coefficients.\nCombined with the McKean–Singer formula it leads to the heat equation proof of the index theorem.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "327NP6S7H5"
    ],
    "kind": "statement",
    "msc": [
      "58J35",
      "58J50",
      "35K08"
    ],
    "name": "Minakshisundaram–Pleijel asymptotic expansion",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Minakshisundaram–Pleijel_zeta_function"
      }
    ],
    "tag": "45P2SEGGR1",
    "uses": [
      "327NP6S7H5",
      "7CBT4SM83Y",
      "ZHX2VYQDZ3",
      "B8TYKV7AA1"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/45P2SEGGR1/",
  "links": [
    {
      "field": "about",
      "target": "327NP6S7H5"
    },
    {
      "field": "uses",
      "target": "327NP6S7H5"
    },
    {
      "field": "uses",
      "target": "7CBT4SM83Y"
    },
    {
      "field": "uses",
      "target": "ZHX2VYQDZ3"
    },
    {
      "field": "uses",
      "target": "B8TYKV7AA1"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/45P2SEGGR1.md",
  "msc": [
    "58J35",
    "58J50",
    "35K08"
  ],
  "name": "Minakshisundaram–Pleijel asymptotic expansion",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/45P/45P2SEGGR1.md",
  "status": null,
  "tag": "45P2SEGGR1",
  "uri": "m2://45P2SEGGR1",
  "url": "https://tamnd.github.io/math2/wiki/minakshisundaram-pleijel-asymptotic-expansion/",
  "writing": {
    "tag": "45P2SEGGR1",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}