{
  "id": "1610.08214",
  "version": "v1",
  "published": "2016-10-26T07:25:35.000Z",
  "updated": "2016-10-26T07:25:35.000Z",
  "title": "Mixed volume preserving flow by powers of homogeneous curvature functions of degree one",
  "authors": [
    "Shunzi Guo"
  ],
  "comment": "33pages. Comments are welcome. arXiv admin note: substantial text overlap with arXiv:1306.4539",
  "categories": [
    "math.DG"
  ],
  "abstract": "This paper concerns the evolution of a closed hypersurface of dimension $n(\\geq 2)$ in the Euclidean space ${\\mathbb{R}}^{n+1}$ under a mixed volume preserving flow. The speed equals a power $\\beta (\\geq 1)$ of homogeneous, either convex or concave, curvature functions of degree one plus a mixed volume preserving term, including the case of powers of the mean curvature and of the Gauss curvature. The main result is that if the initial hypersurface satisfies a suitable pinching condition, there exists a unique, smooth solution of the flow for all times, and the evolving hypersurfaces converge exponentially to a round sphere, enclosing the same mixed volume as the initial hypersurface. This result covers and generalises the previous results for convex hypersurfaces in the Euclidean space by McCoy \\cite{McC05} and Cabezas-Rivas and Sinestrari \\cite{CS10} to more general curvature flows for convex hypersurfaces with similar curvature pinching condition.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-26T07:25:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C44",
      "35K55",
      "58J35",
      "35B40"
    ],
    "keywords": [
      "mixed volume preserving flow",
      "homogeneous curvature functions",
      "euclidean space",
      "convex hypersurfaces",
      "similar curvature pinching condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}