Keywords
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
B. Andrews, Contraction of convex hypersurfaces in Euclidean space, Calc. Var. 2 (1994), 151–171.
B. Andrews, Gauss curvature flow: The fate of the rolling stones, preprint ANU Canberra (1998), pp10.
B. Andrews, Contraction of convex hypersurfaces by their affine normal, J. Diff. Geom. 43 (1996), 207–230.
B. Andrews, Contraction of convex hypersurfaces in Riemannian spaces, J. Diff. Geom. 39 (1994), 407–431.
B. Andrews, Monotone quantities and unique limits for evolving convex hypersurfaces, IMRN 20 (1997), 1001–1031.
B. Andrews, private communication.
S.B. Angenent, J.J.L. Velazques, Degenerate neckpinches in mean curvature flow, J. Reine Angew. Math. 482 (1997), 15–66.
T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampère Equations, Springer-Verlag, New York.
K.A. Brakke, The Motion of a Surface, by its Mean Curvature, Mathematical Notes, Princeton University Press, Princeton.
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/.
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.
B. Chow, Deforming convex hypersurfaces by the nth root, of the Guassian curvature, J. Diff. Geom. 23 (1985), 117–138.
P. T. Chruściel, On Robinson-Trautman Space-Times, Centre for Mathematical Analysis Research Report CMA-R23-90, The Australian National University, Canberra.
D.M. DeTurck, Deforming metrics in direction of their Ricci tensors, J. Diff. Geom. 18 (1983), 157–162.
J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York and London.
K. Ecker, G. Huisken, Interior estimates for hypersurfaces moving by mean curvature, Invent. Math. 105 (1991), 547–569.
K. Ecker, G. Huisken, Mean curvature evolution of entire graphs, Ann. of Math. 130 (1989), 453–471.
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.
L.C. Evans, J. Spruck, Motion of Level Sets by Mean Curvature I, J. Diff. Geom. 33 (1991), 635–681.
W.J. Firey, Shapes of worn stones, Mathematica 21 (1974), 1–11.
Avner F. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, N.J.
M.E. Gage and R.S. Hamilton, The Heat Equation Shrinking Convex Plane Curves, J. Diff Geom. 23, 285–314.
C. Gerhardt, Flow, of nonconvex hypersurfaces into spheres, J. Diff. Geom. 32 (1990), 299–314.
R. Geroch, Energy Extraction, Ann. New York Acad. Sci. 224 (1973), 108–17.
M. Grayson, The heat equation shrinks embedded plane curves to points, J. Diff. Geom. 26 (1987), 285–314.
M. Grayson, Shortening embedded curves, Annals Math. 129 (1989), 71–111.
R.S. Hamilton, The Ricci Flow on Surfaces, Contemporary Mathematics 71, 237–262.
R.S. Hamilton, The formation of singularities in the Ricci Flow, Surveys in Differntial Geometry Vol. II, International Press, Cambridge MA (1993), 7–136.
R.S. Hamilton, Harnack estimate for the mean curvature flow, J. Diff. Geom. 41 (1995), 215–226.
R.S. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom. 17 (1982), 255–306.
R.S. Hamilton, Four manifolds with positive isotropic curvature, Comm. Anal. Geom. 5 (1997), 1–92.
R. S. Hamilton, Isoperimetric estimates for the curve shrinking flow in the plane, Modern Methods in Compl. Anal., Princeton Univ. Press (1992), 201–222.
R. S. Hamilton, CBMS Conference Notes, Hawaii.
G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Diff. Geometry 20 (1984), 237–266.
G. Huisken, Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, Invent. Math. 84 (1986), 463–480.
G. Huisken, Asymptotic behaviour for singularities of the mean curvature flow, J. Diff. Geometry 31 (1990), 285–299.
G. Huisken, Local and global behaviour of hypersurfaces moving by mean curvature, Proceedings of Symposia in Pure Mathematics 54 (1993), 175–191.
G. Huisken, A distance comparison principle for evolving curves, Asian J. Math. 2 (1998), 127–134.
G. Huisken, T. Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality, preprint http://poincare.mathematik.uni-tuebingen.de, to appear.
G. Huisken, T. Ilmanen, The Riemannian Penrose inequality, IMRN 20 (1997), 1045–1058.
G. Huisken, C. Sinestrari, Mean curvature, flow singularities for mean convex surfaces, Calc. Variations, to appear.
G. Huisken, C. Sinestrari, Convexity estimates for mean curvature flow and singularities for mean convex surfaces, preprint, to appear.
T. Ilmanen, Elliptic regularisation and partial regularity for motion by mean curvature, Memoirs AMS 108 (1994), pp90.
J.L. Lions, Sur les problèmes mixtes pour certaines systèmes paraboliques dans des ouverts non cylindriques, Ann. l'Institute Fourier 7, 143–182.
W.W. Mullins, Two-dimensional Motion of Idealised Grain Boundaries, J. Appl. Phys. 27/8 (August 1956), 900–904.
G. Sapiro, A. Tannenbaum, On affine plane curve evolution, J. Funct. Anal. 119 (1994), 79–120.
R. Schoen, L. Simon, S.T. Yau, Curvature estimates for minimal hypersurfaces, Acta Math. 134 (1975), 276–288.
J. Simons, Minimal varicties in Riemannian manifolds, Ann. of Math. 88 (1968), 62–105.
K. Smoczyk, Starshaped hypersurfaces and the mean curvature flow, Preprint (1997), 133pp.
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.
F. Trèves, Relations de domination entre opérateurs différentiels, Acta. Math. 101, 1–139.
K. Tso, Deforming a hypersurface by its Gauss-Kronecker curvature, Comm. Pure Appl. Math. 38 (1985), 867–882.
J. Urbas, On the Expansion of Starshaped Hypersurfaces by Symmetric Functions of Their Principal Curvatures, Math. Z. 205 (1990), 355–372.
B. White, Partial Regularity of Mean Convex Hypersurfaces Flowing by Mean Curvature, IMRN 4 (1994), 185–192.
Editor information
Rights 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