Skip to main content
SpringerLink
Log in

Geometric evolution equations for hypersurfaces

  • Chapter
  • First Online:
Calculus of Variations and Geometric Evolution Problems

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1713))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. B. Andrews, Contraction of convex hypersurfaces in Euclidean space, Calc. Var. 2 (1994), 151–171.

    Article  MathSciNet  MATH  Google Scholar 

  2. B. Andrews, Gauss curvature flow: The fate of the rolling stones, preprint ANU Canberra (1998), pp10.

    Google Scholar 

  3. B. Andrews, Contraction of convex hypersurfaces by their affine normal, J. Diff. Geom. 43 (1996), 207–230.

    MathSciNet  MATH  Google Scholar 

  4. B. Andrews, Contraction of convex hypersurfaces in Riemannian spaces, J. Diff. Geom. 39 (1994), 407–431.

    MathSciNet  MATH  Google Scholar 

  5. B. Andrews, Monotone quantities and unique limits for evolving convex hypersurfaces, IMRN 20 (1997), 1001–1031.

    Article  MathSciNet  MATH  Google Scholar 

  6. B. Andrews, private communication.

    Google Scholar 

  7. S.B. Angenent, J.J.L. Velazques, Degenerate neckpinches in mean curvature flow, J. Reine Angew. Math. 482 (1997), 15–66.

    MathSciNet  MATH  Google Scholar 

  8. T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampère Equations, Springer-Verlag, New York.

    Google Scholar 

  9. K.A. Brakke, The Motion of a Surface, by its Mean Curvature, Mathematical Notes, Princeton University Press, Princeton.

    Google Scholar 

  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. 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.

    MathSciNet  MATH  Google Scholar 

  12. B. Chow, Deforming convex hypersurfaces by the nth root, of the Guassian curvature, J. Diff. Geom. 23 (1985), 117–138.

    MATH  Google Scholar 

  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. D.M. DeTurck, Deforming metrics in direction of their Ricci tensors, J. Diff. Geom. 18 (1983), 157–162.

    MathSciNet  MATH  Google Scholar 

  15. J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York and London.

    Google Scholar 

  16. K. Ecker, G. Huisken, Interior estimates for hypersurfaces moving by mean curvature, Invent. Math. 105 (1991), 547–569.

    Article  MathSciNet  MATH  Google Scholar 

  17. K. Ecker, G. Huisken, Mean curvature evolution of entire graphs, Ann. of Math. 130 (1989), 453–471.

    Article  MathSciNet  MATH  Google Scholar 

  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. L.C. Evans, J. Spruck, Motion of Level Sets by Mean Curvature I, J. Diff. Geom. 33 (1991), 635–681.

    MathSciNet  MATH  Google Scholar 

  20. W.J. Firey, Shapes of worn stones, Mathematica 21 (1974), 1–11.

    MathSciNet  MATH  Google Scholar 

  21. Avner F. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, N.J.

    Google Scholar 

  22. M.E. Gage and R.S. Hamilton, The Heat Equation Shrinking Convex Plane Curves, J. Diff Geom. 23, 285–314.

    Google Scholar 

  23. C. Gerhardt, Flow, of nonconvex hypersurfaces into spheres, J. Diff. Geom. 32 (1990), 299–314.

    MathSciNet  MATH  Google Scholar 

  24. R. Geroch, Energy Extraction, Ann. New York Acad. Sci. 224 (1973), 108–17.

    Article  MATH  Google Scholar 

  25. M. Grayson, The heat equation shrinks embedded plane curves to points, J. Diff. Geom. 26 (1987), 285–314.

    MathSciNet  MATH  Google Scholar 

  26. M. Grayson, Shortening embedded curves, Annals Math. 129 (1989), 71–111.

    Article  MathSciNet  MATH  Google Scholar 

  27. R.S. Hamilton, The Ricci Flow on Surfaces, Contemporary Mathematics 71, 237–262.

    Google Scholar 

  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.

    MATH  Google Scholar 

  29. R.S. Hamilton, Harnack estimate for the mean curvature flow, J. Diff. Geom. 41 (1995), 215–226.

    MathSciNet  MATH  Google Scholar 

  30. R.S. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom. 17 (1982), 255–306.

    MathSciNet  MATH  Google Scholar 

  31. R.S. Hamilton, Four manifolds with positive isotropic curvature, Comm. Anal. Geom. 5 (1997), 1–92.

    Article  MathSciNet  MATH  Google Scholar 

  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. R. S. Hamilton, CBMS Conference Notes, Hawaii.

    Google Scholar 

  34. G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Diff. Geometry 20 (1984), 237–266.

    MathSciNet  MATH  Google Scholar 

  35. G. Huisken, Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, Invent. Math. 84 (1986), 463–480.

    Article  MathSciNet  MATH  Google Scholar 

  36. G. Huisken, Asymptotic behaviour for singularities of the mean curvature flow, J. Diff. Geometry 31 (1990), 285–299.

    MathSciNet  MATH  Google Scholar 

  37. G. Huisken, Local and global behaviour of hypersurfaces moving by mean curvature, Proceedings of Symposia in Pure Mathematics 54 (1993), 175–191.

    Article  MathSciNet  MATH  Google Scholar 

  38. G. Huisken, A distance comparison principle for evolving curves, Asian J. Math. 2 (1998), 127–134.

    Article  MathSciNet  MATH  Google Scholar 

  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. G. Huisken, T. Ilmanen, The Riemannian Penrose inequality, IMRN 20 (1997), 1045–1058.

    Article  MathSciNet  MATH  Google Scholar 

  41. G. Huisken, C. Sinestrari, Mean curvature, flow singularities for mean convex surfaces, Calc. Variations, to appear.

    Google Scholar 

  42. G. Huisken, C. Sinestrari, Convexity estimates for mean curvature flow and singularities for mean convex surfaces, preprint, to appear.

    Google Scholar 

  43. T. Ilmanen, Elliptic regularisation and partial regularity for motion by mean curvature, Memoirs AMS 108 (1994), pp90.

    Article  MathSciNet  MATH  Google Scholar 

  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. W.W. Mullins, Two-dimensional Motion of Idealised Grain Boundaries, J. Appl. Phys. 27/8 (August 1956), 900–904.

    Article  MathSciNet  Google Scholar 

  46. G. Sapiro, A. Tannenbaum, On affine plane curve evolution, J. Funct. Anal. 119 (1994), 79–120.

    Article  MathSciNet  MATH  Google Scholar 

  47. R. Schoen, L. Simon, S.T. Yau, Curvature estimates for minimal hypersurfaces, Acta Math. 134 (1975), 276–288.

    Article  MathSciNet  MATH  Google Scholar 

  48. J. Simons, Minimal varicties in Riemannian manifolds, Ann. of Math. 88 (1968), 62–105.

    Article  MathSciNet  MATH  Google Scholar 

  49. K. Smoczyk, Starshaped hypersurfaces and the mean curvature flow, Preprint (1997), 133pp.

    Google Scholar 

  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. F. Trèves, Relations de domination entre opérateurs différentiels, Acta. Math. 101, 1–139.

    Google Scholar 

  52. K. Tso, Deforming a hypersurface by its Gauss-Kronecker curvature, Comm. Pure Appl. Math. 38 (1985), 867–882.

    Article  MathSciNet  MATH  Google Scholar 

  53. J. Urbas, On the Expansion of Starshaped Hypersurfaces by Symmetric Functions of Their Principal Curvatures, Math. Z. 205 (1990), 355–372.

    Article  MathSciNet  MATH  Google Scholar 

  54. B. White, Partial Regularity of Mean Convex Hypersurfaces Flowing by Mean Curvature, IMRN 4 (1994), 185–192.

    Article  MATH  Google Scholar 

Download references

Authors
  1. Huisken Gerhard
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Polden Alexander
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Stefan Hildebrandt Michael Struwe

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer-Verlag

About this chapter

Cite this chapter

Gerhard, H., Alexander, P. (1999). Geometric evolution equations for hypersurfaces. In: Hildebrandt, S., Struwe, M. (eds) Calculus of Variations and Geometric Evolution Problems. Lecture Notes in Mathematics, vol 1713. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0092669

Download citation

  • DOI: https://doi.org/10.1007/BFb0092669

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-65977-8

  • Online ISBN: 978-3-540-48813-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation