{
  "id": "2103.03336",
  "version": "v1",
  "published": "2021-03-04T21:09:40.000Z",
  "updated": "2021-03-04T21:09:40.000Z",
  "title": "Triangles and triple products of Laplace eigenfunctions",
  "authors": [
    "Emmett L. Wyman"
  ],
  "comment": "27 pages, 1 figure",
  "categories": [
    "math.AP",
    "math.SP"
  ],
  "abstract": "Consider an $L^2$-normalized Laplace-Beltrami eigenfunction $e_\\lambda$ on a compact, boundary-less Riemannian manifold with $\\Delta e_\\lambda = -\\lambda^2 e_\\lambda$. We study eigenfunction triple products \\[ \\langle e_\\lambda e_\\mu, e_\\nu \\rangle = \\int e_\\lambda e_\\mu \\overline{e_\\nu} \\, dV. \\] We show the overall $\\ell^2$-concentration of these triple products is determined by the measure of some set of configurations of triangles with side lengths equal to the frequencies $\\lambda,\\mu,$ and $\\nu$. A rapidly vanishing proportion of this mass lies in the `classically forbidden' regime where $\\lambda, \\mu,$ and $\\nu$ fail to satisfy the triangle inequality. As a consequence, we improve a result by Lu, Sogge, and Steinerberger.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-04T21:09:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58J40",
      "35S30",
      "35Pxx"
    ],
    "keywords": [
      "laplace eigenfunctions",
      "study eigenfunction triple products",
      "side lengths equal",
      "boundary-less riemannian manifold",
      "mass lies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}