{
  "id": "1605.01882",
  "version": "v1",
  "published": "2016-05-06T10:31:29.000Z",
  "updated": "2016-05-06T10:31:29.000Z",
  "title": "The first eigenvalue of the $p$-Laplacian on time dependent Riemannian metrics",
  "authors": [
    "Abimbola Abolarinwa",
    "Jing Mao"
  ],
  "comment": "23 pages",
  "categories": [
    "math.DG"
  ],
  "abstract": "Let $(M,g)$ be an $n$-dimensional compact Riemannian manifold ($n>1$) whose metric $g(t)$ evolves by the generalized abstract geometric flow. This paper discusses the evolution, monotonicity and differentiability for the first eigenvalue of the $p$-Laplacian on $(M,g(t))$ with respect to time evolution. We prove that the first nonzero $p$-eigenvalue is monotone nondecreasing along the flow under certain geometric condition and that the first eigenvalue is differentiable almost everywhere. When $p=2$, we recover the corresponding results for the usual Laplace-Beltrami operator. Our results provide a unified approach to the study of $p$-eigenvalue under various geometric flows",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-06T10:31:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C21",
      "53C44",
      "58C40"
    ],
    "keywords": [
      "time dependent riemannian metrics",
      "first eigenvalue",
      "dimensional compact riemannian manifold",
      "usual laplace-beltrami operator",
      "generalized abstract geometric flow"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}