{
  "id": "2309.16449",
  "version": "v1",
  "published": "2023-09-28T13:59:09.000Z",
  "updated": "2023-09-28T13:59:09.000Z",
  "title": "Mean curvature flows of graphs sliding off to infinity in warped product manifolds",
  "authors": [
    "Naotoshi Fujihara"
  ],
  "comment": "18 pages",
  "categories": [
    "math.DG"
  ],
  "abstract": "We study mean curvature flows in a warped product manifold defined by a closed Riemannian manifold and $\\mathbb{R}$. In such a warped product manifold, we can define the notion of a graph, called a geodesic graph. We prove that the curve shortening flow preserves a geodesic graph for any warping function, and the mean curvature flow of hypersurfaces preserves a geodesic graph for some monotone convex warping functions. In particular, we consider some warping functions that go to zero at infinity, which means that the curves or hypersurfaces go to a point at infinity along the flow. In such a case, we prove the long-time existence of the flow and that the curvature and its higher-order derivatives go to zero along the flow.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-28T13:59:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C44"
    ],
    "keywords": [
      "warped product manifold",
      "geodesic graph",
      "graphs sliding",
      "study mean curvature flows",
      "curve shortening flow preserves"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}