# Introduction to mean curvature flow

Roberta Alessandroni^{[1]}

- [1] Albert-Einstein-Institut Max-Planck-Institut für Gravitationsphysik Am Mühlenberg 1 14476 Golm (Germany)

Séminaire de théorie spectrale et géométrie (2008-2009)

- Volume: 27, page 1-9
- ISSN: 1624-5458

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top## Abstract

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topAlessandroni, Roberta. "Introduction to mean curvature flow." Séminaire de théorie spectrale et géométrie 27 (2008-2009): 1-9. <http://eudml.org/doc/116457>.

@article{Alessandroni2008-2009,

abstract = {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.},

affiliation = {Albert-Einstein-Institut Max-Planck-Institut für Gravitationsphysik Am Mühlenberg 1 14476 Golm (Germany)},

author = {Alessandroni, Roberta},

journal = {Séminaire de théorie spectrale et géométrie},

keywords = {mean curvature flow; curve shortening flow; mean curvature flow for graphs},

language = {eng},

pages = {1-9},

publisher = {Institut Fourier},

title = {Introduction to mean curvature flow},

url = {http://eudml.org/doc/116457},

volume = {27},

year = {2008-2009},

}

TY - JOUR

AU - Alessandroni, Roberta

TI - Introduction to mean curvature flow

JO - Séminaire de théorie spectrale et géométrie

PY - 2008-2009

PB - Institut Fourier

VL - 27

SP - 1

EP - 9

AB - 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.

LA - eng

KW - mean curvature flow; curve shortening flow; mean curvature flow for graphs

UR - http://eudml.org/doc/116457

ER -

## References

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- Klaus Ecker, Gerhard Huisken, Interior estimates for hypersurfaces moving by mean curvature, Invent. Math. 105 (1991), 547-569 Zbl0707.53008MR1117150
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- Matthew A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom. 26 (1987), 285-314 Zbl0667.53001MR906392
- Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982), 255-306 Zbl0504.53034MR664497
- Gerhard Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), 237-266 Zbl0556.53001MR772132
- Gerhard Huisken, Carlo Sinestrari, Mean curvature flow with surgeries of two-convex hypersurfaces, Invent. Math. 175 (2009), 137-221 Zbl1170.53042MR2461428

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