|
|
| |
| On Asymptotic Behavior for Singularities of the Powers of Mean Curvature Flow |
| |
Citation： |
Weimin SHENG,Chao WU.On Asymptotic Behavior for Singularities of the Powers of Mean Curvature Flow[J].Chinese Annals of Mathematics B,2009,(1):51~66 |
| Page view： 3019
Net amount： 2243 |
Authors： |
Weimin SHENG; Chao WU; |
Foundation： |
the National Natural Science Foundation of China (Nos. 10771189, 10831008). |
|
|
| Abstract： |
Let $M^n$ be a smooth, compact manifold without boundary, and
$F_0:M^n\to R^{n+1}$ a smooth immersion which is convex. The
one-parameter families $F(\,\cdot\,,t):M^n\times[0,T)\to R^{n+1}$ of
hypersurfaces $M_t^n=F(\,\cdot\,,t)(M^n)$ satisfy an initial value
problem $ \frac{\d F}{\d t}(\,\cdot\,,t)
=-H^k(\,\cdot\,,t)\nu(\,\cdot\,,t)$,
$F(\,\cdot\,,0)=F_0(\,\cdot\,)$, where $H$ is the mean curvature and
$\nu(\,\cdot\,,t)$ is the outer unit normal at $F(\,\cdot\,,t)$,
such that $-H\nu=\overrightarrow H$ is the mean curvature vector,
and $k>0$ is a constant. This problem is called $H^k$-flow. Such
flow will develop singularities after finite time. According to the
blow-up rate of the square norm of the second fundamental forms, the
authors analyze the structure of the rescaled limit by classifying
the singularities as two types, i.e., Type I and Type II. It is
proved that for Type I singularity, the limiting hypersurface
satisfies an elliptic equation; for Type II singularity, the
limiting hypersurface must be a translating soliton. |
Keywords： |
$H^k$-Curvature flow, Type I singularities, Type II singularities |
Classification： |
53C44, 35K55 |
|
|
Download PDF Full-Text
|
|
|
|
