This is a short overview on the most classical results on mean curvature flow as a flow of smooth hypersurfaces. First of all we define the mean curvature flow as a quasilinear parabolic equation and give some easy examples of evolution. Then we consider the M.C.F. on convex surfaces and sketch the proof of the convergence to a round point. Some interesting results on the M.C.F. for entire graphs are also mentioned. In particular when we consider the case of dimension one, we can compute the equation for the translating graph solution to the curve shortening flow and solve it directly.
@article{TSG_2008-2009__27__1_0, author = {Roberta Alessandroni}, title = {Introduction to mean curvature flow}, journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie}, pages = {1--9}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {27}, year = {2008-2009}, doi = {10.5802/tsg.267}, mrnumber = {2799143}, language = {en}, url = {https://proceedings.centre-mersenne.org/articles/10.5802/tsg.267/} }
TY - JOUR AU - Roberta Alessandroni TI - Introduction to mean curvature flow JO - Séminaire de théorie spectrale et géométrie PY - 2008-2009 SP - 1 EP - 9 VL - 27 PB - Institut Fourier PP - Grenoble UR - https://proceedings.centre-mersenne.org/articles/10.5802/tsg.267/ DO - 10.5802/tsg.267 LA - en ID - TSG_2008-2009__27__1_0 ER -
%0 Journal Article %A Roberta Alessandroni %T Introduction to mean curvature flow %J Séminaire de théorie spectrale et géométrie %D 2008-2009 %P 1-9 %V 27 %I Institut Fourier %C Grenoble %U https://proceedings.centre-mersenne.org/articles/10.5802/tsg.267/ %R 10.5802/tsg.267 %G en %F TSG_2008-2009__27__1_0
Roberta Alessandroni. Introduction to mean curvature flow. Séminaire de théorie spectrale et géométrie, Volume 27 (2008-2009), pp. 1-9. doi : 10.5802/tsg.267. https://proceedings.centre-mersenne.org/articles/10.5802/tsg.267/
[1] Klaus Ecker Regularity theory for mean curvature flow, Progress in Nonlinear Differential Equations and their Applications, 57, Birkhäuser Boston Inc., Boston, MA, 2004 | MR | Zbl
[2] Klaus Ecker; Gerhard Huisken Mean curvature evolution of entire graphs, Ann. of Math. (2), Volume 130 (1989) no. 3, pp. 453-471 | MR | Zbl
[3] Klaus Ecker; Gerhard Huisken Interior estimates for hypersurfaces moving by mean curvature, Invent. Math., Volume 105 (1991) no. 3, pp. 547-569 | MR | Zbl
[4] M. E. Gage Curve shortening makes convex curves circular, Invent. Math., Volume 76 (1984) no. 2, pp. 357-364 | MR | Zbl
[5] Matthew A. Grayson The heat equation shrinks embedded plane curves to round points, J. Differential Geom., Volume 26 (1987) no. 2, pp. 285-314 | MR | Zbl
[6] Richard S. Hamilton Three-manifolds with positive Ricci curvature, J. Differential Geom., Volume 17 (1982) no. 2, pp. 255-306 | MR | Zbl
[7] Gerhard Huisken Flow by mean curvature of convex surfaces into spheres, J. Differential Geom., Volume 20 (1984) no. 1, pp. 237-266 | MR | Zbl
[8] Gerhard Huisken; Carlo Sinestrari Mean curvature flow with surgeries of two-convex hypersurfaces, Invent. Math., Volume 175 (2009) no. 1, pp. 137-221 | MR | Zbl
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