{
  "id": "1604.01694",
  "version": "v1",
  "published": "2016-04-06T17:07:52.000Z",
  "updated": "2016-04-06T17:07:52.000Z",
  "title": "On the classification of ancient solutions to curvature flows on the sphere",
  "authors": [
    "Paul Bryan",
    "Mohammad N. Ivaki",
    "Julian Scheuer"
  ],
  "comment": "20 pages, 1 figure. Comments are welcome",
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "We consider the evolution of hypersurfaces on the unit sphere $\\mathbb{S}^{n+1}$ by smooth functions of the Weingarten map. We introduce the notion of `quasi-ancient' solutions for flows that do not admit non-trivial, convex, ancient solutions. Such solutions are somewhat analogous to ancient solutions for flows such as the mean curvature flow, or 1-homogeneous flows. The techniques presented here allow us to prove that any convex, quasi-ancient solution of a curvature flow which satisfies a backwards in time uniform bound on mean curvature must be stationary or a family of shrinking geodesic spheres. The main tools are geometric, employing the maximum principle, a rigidity result in the sphere and an Alexandrov reflection argument. We emphasize that no homogeneity or convexity/concavity restrictions are placed on the speed, though we do also offer a short classification proof for several such restricted cases.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-06T17:07:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ancient solutions",
      "mean curvature flow",
      "alexandrov reflection argument",
      "short classification proof",
      "time uniform bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}