Formazione di singolarità nel moto per curvatura media

Carlo Sinestrari

Bollettino dell'Unione Matematica Italiana (2001)

  • Volume: 4-B, Issue: 1, page 107-119
  • ISSN: 0392-4041

Abstract

top
We study the formation of singularities for hypersurfaces evolving by mean curvature. After recalling the basic properties of the flow and the classical results about curves and convex surfaces, we give account of some recent developments of the theory for the case of surfaces with positive mean curvature. We show that for such surfaces we can obtain a–priori estimates on the principal curvatures which enable us to classify the singular profiles by the use of rescaling techniques.

How to cite

top

Sinestrari, Carlo. "Formazione di singolarità nel moto per curvatura media." Bollettino dell'Unione Matematica Italiana 4-B.1 (2001): 107-119. <http://eudml.org/doc/195798>.

@article{Sinestrari2001,
author = {Sinestrari, Carlo},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {ita},
month = {2},
number = {1},
pages = {107-119},
publisher = {Unione Matematica Italiana},
title = {Formazione di singolarità nel moto per curvatura media},
url = {http://eudml.org/doc/195798},
volume = {4-B},
year = {2001},
}

TY - JOUR
AU - Sinestrari, Carlo
TI - Formazione di singolarità nel moto per curvatura media
JO - Bollettino dell'Unione Matematica Italiana
DA - 2001/2//
PB - Unione Matematica Italiana
VL - 4-B
IS - 1
SP - 107
EP - 119
LA - ita
UR - http://eudml.org/doc/195798
ER -

References

top
  1. ABRESCH, U.- LANGER, J., The normalized curve shortening flow and homothetic solutions, J. Differential Geom., 23 (1986), 175-196. Zbl0592.53002MR845704
  2. ALTSCHULER, S.- ANGENENT, S. B.- GIGA, Y., Mean curvature flow through singularities for surfaces of rotation, J. Geom. Analysis, 5 (1995), 293-358. Zbl0847.58072MR1360824
  3. AMBROSIO, L., Geometric evolution problems, distance function and viscosity solutions, in «Calculus of variations and partial differential equations (Pisa, 1996)» (G. Buttazzo, A. Marino, M.K.V. Murthy eds.) Springer. Berlin, (2000). Zbl0956.35002MR1757696
  4. ANDREWS, B., Contraction of convex hypersurfaces in Riemannian spaces, J. Differential Geom., 39 (1994), 407-431. Zbl0797.53044MR1267897
  5. ANGENENT, S. B., On the formation of singularities in the curve shortening flow, J. Differential Geom., 33 (1991), 601-633. Zbl0731.53002MR1100205
  6. ANGENENT, S. B., Some recent results on mean curvature flow, in «Recent advances in partial differential equations (El Escorial, 1992)», RAM Res. Appl. Math., 30, Masson, Paris, 1994. Zbl0796.35068MR1266199
  7. ANGENENT, S. B.- VELAZQUEZ, J. J. L., Asymptotic shape of cusp singularities in curve shortening, Duke Math. J., 77, no. 1 (1995), 71-110. Zbl0829.35058MR1317628
  8. ANGENENT, S. B.- VELAZQUEZ, J. J. L., Degenerate neckpinches in mean curvature flow, J. Reine Angew. Math., 482 (1997), 15-66. Zbl0866.58055MR1427656
  9. BRAKKE, K. A., The motion of a surface by its mean curvature, Princeton University Press, Princeton (1978). Zbl0386.53047MR485012
  10. CHEN, Y. G.- GIGA, Y.- GOTO, S., Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., 33 (1991), 749-786. Zbl0696.35087MR1100211
  11. EELLS, J.- SAMPSON, J. H., Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964), 109-160. Zbl0122.40102MR164306
  12. EVANS, L. C.- SPRUCK, J., Motion of level sets by mean curvature, I, J. Differential Geom., 33 (1991), 635-681. Zbl0726.53029MR1100206
  13. EVANS, L. C.- SPRUCK, J., Motion of level sets by mean curvature, II, Trans. Amer. Math. Soc., 330 (1992), 321-332. Zbl0776.53005MR1068927
  14. GAGE, M.- HAMILTON, R. S., The heat equation shrinking convex plane curves, J. Differential Geom., 23 (1986), 69-96. Zbl0621.53001MR840401
  15. GRAYSON, M. A., The heat equation shrinks embedded plane curves to round points, J. Differential Geom., 26 (1987), 285-314. Zbl0667.53001MR906392
  16. GRAYSON, M. A., A short note on the evolution of a surface by its mean curvature, Duke Math. J., 58 (1989), 555-558. Zbl0677.53059MR1016434
  17. HAMILTON, R. S., Three-manifolds with positive Ricci curvature, J. Differential Geom., 17 (1982), 255-306. Zbl0504.53034MR664497
  18. HAMILTON, R. S., Harnack estimate for the mean curvature flow, J. Differential Geom., 41 (1995), 215-226. Zbl0827.53006MR1316556
  19. HAMILTON, R. S., Four-manifolds with positive isotropic curvature, Comm. Anal. Geom., 5 (1997), 1-92. Zbl0892.53018MR1456308
  20. HUISKEN, G., Flow by mean curvature of convex surfaces into spheres, J. Differential Geom., 20 (1984), 237-266. Zbl0556.53001MR772132
  21. HUISKEN, G., Asymptotic behaviour for singularities of the mean curvature flow, J. Differential Geom., 31 (1990), 285-299. Zbl0694.53005MR1030675
  22. HUISKEN, G., Local and global behaviour of hypersurfaces moving by mean curvature, Proceedings of Symposia in Pure Mathematics, 54 (1993), 175-191. Zbl0791.58090MR1216584
  23. HUISKEN, G., Evolution of hypersurfaces by their curvature in Riemannian manifolds. Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998). Doc. Math. (1998), 349-360. Zbl0912.53038MR1648085
  24. HUISKEN, G.- ILMANEN, T., The inverse mean curvature flow and the Riemannian Penrose inequality, in corso di stampa su J. Differential Geom. Zbl1055.53052MR1916951
  25. HUISKEN, G.- SINESTRARI, C., Mean curvature flow singularities for mean convex surfaces, Calc. Variations, 8 (1999), 1-14. Zbl0992.53052MR1666878
  26. HUISKEN, G.- SINESTRARI, C., Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math., 183 (1999), 45-70. Zbl0992.53051MR1719551
  27. ILMANEN, T., Elliptic regularization and partial regularity for motion by mean curvature, Mem. Amer. Math. Soc., 108 (1994). Zbl0798.35066MR1196160
  28. SOUGANIDIS, P. E., Front propagation: theory and applications in: «Viscosity solutions and applications» (I. Capuzzo Dolcetta and P.L. Lions, Eds.), Springer-Verlag, Berlin (1997), 186-242. Zbl0882.35016MR1462703
  29. WHITE, B.The nature of singularities in mean curvature flow of mean-convex sets, preprint (1998). Zbl1027.53078MR1937202

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.