{
  "id": "1407.0864",
  "version": "v1",
  "published": "2014-07-03T11:01:35.000Z",
  "updated": "2014-07-03T11:01:35.000Z",
  "title": "Monotonicity of the first Dirichlet eigenvalue of the Laplacian on manifolds of nonpositive curvature",
  "authors": [
    "Tom Carroll",
    "Jesse Ratzkin"
  ],
  "comment": "18 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $(M,g)$ be a complete manifold of nonpositive scalar curvature, let $\\Omega\\subset M$ be a suitable domain, and let $\\lambda(\\Omega)$ be the first Dirichlet eigenvalue of the Laplace-Beltrami operator on $\\Omega$. We prove several bounds for the rate of decrease of $\\lambda(\\Omega)$ and $\\Omega$ increases, and a result comparing the rate of decrease of $\\lambda$ before and after a conformal diffeomorphism. Along the way, we prove a reverse-Holder inequality for the first eigenfunction, which generalizes results of Chiti to the monifold setting and may be of independent interest",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-03T11:01:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P15"
    ],
    "keywords": [
      "first dirichlet eigenvalue",
      "nonpositive curvature",
      "monotonicity",
      "complete manifold",
      "generalizes results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.0864C"
    }
  }
}