# Convexity estimates for flows by powers of the mean curvature

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

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

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topSchulze, 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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- [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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