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Geometric evolution equations for hypersurfaces

  • Gerhard Huisken 
  • Alexander Polden 
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 1713)

Keywords

Riemannian Manifold Curvature Flow Fundamental Form Principal Curvature Ricci Flow 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    B. Andrews, Contraction of convex hypersurfaces in Euclidean space, Calc. Var. 2 (1994), 151–171.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    B. Andrews, Gauss curvature flow: The fate of the rolling stones, preprint ANU Canberra (1998), pp10.Google Scholar
  3. [3]
    B. Andrews, Contraction of convex hypersurfaces by their affine normal, J. Diff. Geom. 43 (1996), 207–230.MathSciNetzbMATHGoogle Scholar
  4. [4]
    B. Andrews, Contraction of convex hypersurfaces in Riemannian spaces, J. Diff. Geom. 39 (1994), 407–431.MathSciNetzbMATHGoogle Scholar
  5. [5]
    B. Andrews, Monotone quantities and unique limits for evolving convex hypersurfaces, IMRN 20 (1997), 1001–1031.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    B. Andrews, private communication.Google Scholar
  7. [7]
    S.B. Angenent, J.J.L. Velazques, Degenerate neckpinches in mean curvature flow, J. Reine Angew. Math. 482 (1997), 15–66.MathSciNetzbMATHGoogle Scholar
  8. [8]
    T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampère Equations, Springer-Verlag, New York.Google Scholar
  9. [9]
    K.A. Brakke, The Motion of a Surface, by its Mean Curvature, Mathematical Notes, Princeton University Press, Princeton.Google Scholar
  10. [10]
    J. W. Cahn, W. C. Carter, A. R. Roosen and J. E. Taylor, Shape Evolution by Surface Diffusion and Surface Attachment Limited Kinetics on Completely Faceted Surfaces available over http://www.ctcms.nist.gov/≈roosen/SD_SALK/.Google Scholar
  11. [11]
    Y.G. Chen, Y. Giga, and S. Goto, Uniqueness and Existence of Viscosity Solutions of Generalized Mean Curvature Flow Equations, J Diff. Geom. 33 (1991), 749–786.MathSciNetzbMATHGoogle Scholar
  12. [12]
    B. Chow, Deforming convex hypersurfaces by the nth root, of the Guassian curvature, J. Diff. Geom. 23 (1985), 117–138.zbMATHGoogle Scholar
  13. [13]
    P. T. Chruściel, On Robinson-Trautman Space-Times, Centre for Mathematical Analysis Research Report CMA-R23-90, The Australian National University, Canberra.Google Scholar
  14. [14]
    D.M. DeTurck, Deforming metrics in direction of their Ricci tensors, J. Diff. Geom. 18 (1983), 157–162.MathSciNetzbMATHGoogle Scholar
  15. [15]
    J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York and London.Google Scholar
  16. [16]
    K. Ecker, G. Huisken, Interior estimates for hypersurfaces moving by mean curvature, Invent. Math. 105 (1991), 547–569.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    K. Ecker, G. Huisken, Mean curvature evolution of entire graphs, Ann. of Math. 130 (1989), 453–471.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    S.D. Eidel'man, Parabolic Equations, in Partial Differential Equations VI, Encyclopaedia of Mathematical Sciences, Volume 63, editors Yu.V. Egerov and M.A. Shubin, Springer-Verlag, Berlin, Heidelberg, New York.Google Scholar
  19. [19]
    L.C. Evans, J. Spruck, Motion of Level Sets by Mean Curvature I, J. Diff. Geom. 33 (1991), 635–681.MathSciNetzbMATHGoogle Scholar
  20. [20]
    W.J. Firey, Shapes of worn stones, Mathematica 21 (1974), 1–11.MathSciNetzbMATHGoogle Scholar
  21. [21]
    Avner F. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, N.J.Google Scholar
  22. [22]
    M.E. Gage and R.S. Hamilton, The Heat Equation Shrinking Convex Plane Curves, J. Diff Geom. 23, 285–314.Google Scholar
  23. [23]
    C. Gerhardt, Flow, of nonconvex hypersurfaces into spheres, J. Diff. Geom. 32 (1990), 299–314.MathSciNetzbMATHGoogle Scholar
  24. [24]
    R. Geroch, Energy Extraction, Ann. New York Acad. Sci. 224 (1973), 108–17.CrossRefzbMATHGoogle Scholar
  25. [25]
    M. Grayson, The heat equation shrinks embedded plane curves to points, J. Diff. Geom. 26 (1987), 285–314.MathSciNetzbMATHGoogle Scholar
  26. [26]
    M. Grayson, Shortening embedded curves, Annals Math. 129 (1989), 71–111.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    R.S. Hamilton, The Ricci Flow on Surfaces, Contemporary Mathematics 71, 237–262.Google Scholar
  28. [28]
    R.S. Hamilton, The formation of singularities in the Ricci Flow, Surveys in Differntial Geometry Vol. II, International Press, Cambridge MA (1993), 7–136.zbMATHGoogle Scholar
  29. [29]
    R.S. Hamilton, Harnack estimate for the mean curvature flow, J. Diff. Geom. 41 (1995), 215–226.MathSciNetzbMATHGoogle Scholar
  30. [30]
    R.S. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom. 17 (1982), 255–306.MathSciNetzbMATHGoogle Scholar
  31. [31]
    R.S. Hamilton, Four manifolds with positive isotropic curvature, Comm. Anal. Geom. 5 (1997), 1–92.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    R. S. Hamilton, Isoperimetric estimates for the curve shrinking flow in the plane, Modern Methods in Compl. Anal., Princeton Univ. Press (1992), 201–222.Google Scholar
  33. [33]
    R. S. Hamilton, CBMS Conference Notes, Hawaii.Google Scholar
  34. [34]
    G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Diff. Geometry 20 (1984), 237–266.MathSciNetzbMATHGoogle Scholar
  35. [35]
    G. Huisken, Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, Invent. Math. 84 (1986), 463–480.MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    G. Huisken, Asymptotic behaviour for singularities of the mean curvature flow, J. Diff. Geometry 31 (1990), 285–299.MathSciNetzbMATHGoogle Scholar
  37. [37]
    G. Huisken, Local and global behaviour of hypersurfaces moving by mean curvature, Proceedings of Symposia in Pure Mathematics 54 (1993), 175–191.MathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    G. Huisken, A distance comparison principle for evolving curves, Asian J. Math. 2 (1998), 127–134.MathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    G. Huisken, T. Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality, preprint http://poincare.mathematik.uni-tuebingen.de, to appear.Google Scholar
  40. [40]
    G. Huisken, T. Ilmanen, The Riemannian Penrose inequality, IMRN 20 (1997), 1045–1058.MathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    G. Huisken, C. Sinestrari, Mean curvature, flow singularities for mean convex surfaces, Calc. Variations, to appear.Google Scholar
  42. [42]
    G. Huisken, C. Sinestrari, Convexity estimates for mean curvature flow and singularities for mean convex surfaces, preprint, to appear.Google Scholar
  43. [43]
    T. Ilmanen, Elliptic regularisation and partial regularity for motion by mean curvature, Memoirs AMS 108 (1994), pp90.MathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    J.L. Lions, Sur les problèmes mixtes pour certaines systèmes paraboliques dans des ouverts non cylindriques, Ann. l'Institute Fourier 7, 143–182.Google Scholar
  45. [45]
    W.W. Mullins, Two-dimensional Motion of Idealised Grain Boundaries, J. Appl. Phys. 27/8 (August 1956), 900–904.MathSciNetCrossRefGoogle Scholar
  46. [46]
    G. Sapiro, A. Tannenbaum, On affine plane curve evolution, J. Funct. Anal. 119 (1994), 79–120.MathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    R. Schoen, L. Simon, S.T. Yau, Curvature estimates for minimal hypersurfaces, Acta Math. 134 (1975), 276–288.MathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    J. Simons, Minimal varicties in Riemannian manifolds, Ann. of Math. 88 (1968), 62–105.MathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    K. Smoczyk, Starshaped hypersurfaces and the mean curvature flow, Preprint (1997), 133pp.Google Scholar
  50. [50]
    V. A. Solonnikov, Green matrices for parabolic boundary value problems, Zap. Nauch. Sem. Leningr. Otd. Math. Inst. Steklova 14 256–287, translated in Semin. Math. Steklova Math. Inst. Leningrad (1972), 109–121.Google Scholar
  51. [51]
    F. Trèves, Relations de domination entre opérateurs différentiels, Acta. Math. 101, 1–139.Google Scholar
  52. [52]
    K. Tso, Deforming a hypersurface by its Gauss-Kronecker curvature, Comm. Pure Appl. Math. 38 (1985), 867–882.MathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    J. Urbas, On the Expansion of Starshaped Hypersurfaces by Symmetric Functions of Their Principal Curvatures, Math. Z. 205 (1990), 355–372.MathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    B. White, Partial Regularity of Mean Convex Hypersurfaces Flowing by Mean Curvature, IMRN 4 (1994), 185–192.CrossRefzbMATHGoogle Scholar

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© Springer-Verlag 1999

Authors and Affiliations

  • Gerhard Huisken 
  • Alexander Polden 

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