# MathSciDoc: An Archive for Mathematician ∫

#### Differential Geometrymathscidoc:1609.10341

JDG, 100, (2), 301-347, 2015
We consider contracting and expanding curvature flows in \$\Ss\$. When the flow hypersurfaces are strictly convex we establish a relation between the contracting hypersurfaces and the expanding hypersurfaces which is given by the Gau{\ss} map. The contracting hypersurfaces shrink to a point \$x_0\$ while the expanding hypersurfaces converge to the equator of the hemisphere \$\mc H(-x_0)\$. After rescaling, by the same scale factor, the rescaled hypersurfaces converge to the unit spheres with centers \$x_0\$ \resp \$-x_0\$ exponentially fast in \$C^\un(\Ss[n])\$.
curvature flows, inverse curvature flows, contracting curvature flows, sphere, polar sets, dual flows, elementary symmetric polynomials
```@inproceedings{claus2015curvature,
title={Curvature flows in the sphere},
author={Claus Gerhardt},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160919234058831773019},
booktitle={JDG},
volume={100},
number={2},
pages={301-347},
year={2015},
}
```
Claus Gerhardt. Curvature flows in the sphere. 2015. Vol. 100. In JDG. pp.301-347. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160919234058831773019.