{
  "id": "1406.1080",
  "version": "v2",
  "published": "2014-06-04T15:44:58.000Z",
  "updated": "2014-10-08T10:01:24.000Z",
  "title": "On largeness and multiplicity of the first eigenvalue of hyperbolic surfaces",
  "authors": [
    "Sugata Mondal"
  ],
  "comment": "16 pages, 2 figures",
  "categories": [
    "math.DG"
  ],
  "abstract": "We apply topological methods to study the smallest non-zero number $\\lambda_1$ in the spectrum of the Laplacian on finite area hyperbolic surfaces. For closed hyperbolic surfaces of genus two we show that the set $\\{S \\in {\\mathcal{M}_2}: {\\lambda_1}(S) > 1/4 \\}$ is unbounded and disconnects the moduli space ${\\mathcal{M}_2}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-04T15:44:58.000Z",
      "abstract": "The smallest non-zero number in the spectrum of the Laplace operator on a smooth surface $S$ of finite area is denoted by ${\\lambda_1}(S)$. The question of existence of closed (finite area) hyperbolic surfaces with $\\lambda_1$ at least $\\frac{1}{4}$ dates back to the paper \\cite{Se} of Atle Selberg where he conjectured that for any congruence subgroup $\\Gamma$ of SL$(2, \\mathbb{Z})$, ${\\lambda_1}(\\mathbb{H}/\\Gamma) \\geq \\frac{1}{4}$. It has been extensively studied in the literature (see for example \\cite{B1} \\cite{B2}, \\cite{BBD}, \\cite{B-M}) providing a satisfactory but not quite complete answer for surfaces of large genus (asymptotic behavior). For example, it is has been achieved that $\\lambda_1$ can get close to $\\frac{3}{16}$, or even $\\frac{1}{4}$ if Selberg's conjecture is true, but possibly only from below. The first result of this paper shows that the set $\\{ S \\in {\\mathcal{M}_2}: {\\lambda_1}(S) \\geq \\frac{1}{4} \\}$ disconnects the moduli space ${\\mathcal{M}_2}$ of closed hyperbolic surfaces of genus two. Using this fact we then show the existence of {\\it branches of eigenvalues} in $\\mathcal{M}_g$ (for any $g \\geq 3$) that start as $\\lambda_1$ and eventually becomes bigger than $\\frac{1}{4}$. Later in the paper we concentrate on one of the main problems of carrying over the method for genus two to higher genus, the multiplicity of $\\lambda_1$. In \\S4 we show that for punctured spheres this multiplicity is at most two independent of the number of punctures.",
      "comment": "15 pages, 4 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-10-08T10:01:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P05",
      "58G20"
    ],
    "keywords": [
      "first eigenvalue",
      "multiplicity",
      "finite area hyperbolic surfaces",
      "smallest non-zero number",
      "moduli space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.1080M"
    }
  }
}