Convexity estimates for flows by powers of the mean curvature

Felix Schulze

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2006)

  • Volume: 5, Issue: 2, page 261-277
  • ISSN: 0391-173X

Abstract

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We study the evolution of a closed, convex hypersurface in ℝ n + 1 in direction of its normal vector, where the speed equals a power k ≥ 1 of the mean curvature. We show that if initially the ratio of the biggest and smallest principal curvatures at every point is close enough to 1 , depending only on k and n , then this is maintained under the flow. As a consequence we obtain that, when rescaling appropriately as the flow contracts to a point, the evolving surfaces converge to the unit sphere.

How to cite

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Schulze, Felix. "Convexity estimates for flows by powers of the mean curvature." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 5.2 (2006): 261-277. <http://eudml.org/doc/240152>.

@article{Schulze2006,
abstract = {We study the evolution of a closed, convex hypersurface in $\mathbb \{R\}^\{n+1\}$ in direction of its normal vector, where the speed equals a power $k\ge 1$ of the mean curvature. We show that if initially the ratio of the biggest and smallest principal curvatures at every point is close enough to $1$, depending only on $k$ and $n$, then this is maintained under the flow. As a consequence we obtain that, when rescaling appropriately as the flow contracts to a point, the evolving surfaces converge to the unit sphere.},
author = {Schulze, Felix},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {2},
pages = {261-277},
publisher = {Scuola Normale Superiore, Pisa},
title = {Convexity estimates for flows by powers of the mean curvature},
url = {http://eudml.org/doc/240152},
volume = {5},
year = {2006},
}

TY - JOUR
AU - Schulze, Felix
TI - Convexity estimates for flows by powers of the mean curvature
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2006
PB - Scuola Normale Superiore, Pisa
VL - 5
IS - 2
SP - 261
EP - 277
AB - We study the evolution of a closed, convex hypersurface in $\mathbb {R}^{n+1}$ in direction of its normal vector, where the speed equals a power $k\ge 1$ of the mean curvature. We show that if initially the ratio of the biggest and smallest principal curvatures at every point is close enough to $1$, depending only on $k$ and $n$, then this is maintained under the flow. As a consequence we obtain that, when rescaling appropriately as the flow contracts to a point, the evolving surfaces converge to the unit sphere.
LA - eng
UR - http://eudml.org/doc/240152
ER -

References

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  1. [1] B. Andrews, Contraction of convex hypersurfaces in Euclidian space, Calc. Var. Partial Differential Equations 2 (1994), 151–171. Zbl0805.35048MR1385524
  2. [2] B. Andrews, Gauss curvature flow: the fate of the rolling stones, Invent. Math. 138 (1999), 151–161. Zbl0936.35080MR1714339
  3. [3] B. Andrews, Moving surfaces by non-concave curvature functions, 2004, arXiv:math.DG/0402273. Zbl1203.53062MR2729317
  4. [4] B. Chow, Deforming convex hypersurfaces by the n th root of the Gaussian curvature, J. Differential Geom. 22 (1985), 117–138. Zbl0589.53005MR826427
  5. [5] B. Chow, Deforming hypersurfaces by the square root of the scalar curvature, Invent. Math. 87 (1987), 63–82. Zbl0608.53005MR862712
  6. [6] E. DiBenedetto and A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math. 357 (1985), 1–22. Zbl0549.35061MR783531
  7. [7] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), 237–266. Zbl0556.53001MR772132
  8. [8] O. C. Schnürer, Surfaces contracting with speed | A | 2 , J. Differential Geom. 71 (2005), 347–363. Zbl1101.53002MR2198805
  9. [9] F. Schulze, Evolution of convex hypersurfaces by powers of the mean curvature, Math. Z. 251 (2005), 721–733. Zbl1087.53062MR2190140

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