##### Department of Mathematics,

University of California San Diego

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### Differential Geometry Seminar

## Liang Zhao

#### UCSD

## An unsolved problem about the self-shrinker in the mean curvature flow

##### Abstract:

The problem says that if $M$ is a smooth complete embedded self-shrinker with polynomial volume growth in Euclidean space and the squared norm of the second fundamental form $|A|^2 =$ constant, then $M$ is a generalized cylinder. It has been verified in dimension 2 without the assumption of polynomial volume growth. Cao and Li had proved if $M$ is an n-dimensional complete self-shrinker with polynomial volume growth in $R^n+q$, and if $|A|^2 \leq 1$, then $M$ is must be one of the generalize cylinders. But for the case $|A|^2 >1$, they don't know what it is. Therefore, Qingming Cheng and Guoxin Wei proved if the squared norm of the second fundamental form $|A|^2$ is constant and $|A|^2 \leq 10/7$, then $M$ is must be one of the generalize cylinders. So we guess that it may be true if the squared norm of the second fundamental form $|A|^2$ is constant. This will be a continuation of the talk given on August 26th.

Organizer: Lei Ni

### September 2, 2016

### 10:00 AM

### AP&M 5829

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