# Postdoc Seminar

## Conformal metrics on the unit ball with prescribed mean curvature

### 张宏 博士 (新加坡国立大学)

#### 2014年7月8日(周二)下午 2:00-3:00 闵行校区行政楼12楼PDE中心1202报告厅

Abstract.This talk focuses on the prescribing mean curvature problem on the unit ball in the Euclidean space with dimension three or higher. Such problem is well known and attracts a lot of attention. If the candidate $f$ for the prescribed mean curvature is sufficiently close to the mean curvature of the standard metric in the infinity norm, then the existence of solution has been known for more than fifteen years. It is interesting to investigate how large that difference can be. More precisely, we assume that the given candidate $f$ is a smooth positive Morse function which is non-degenerate in the sense that $|\nabla f|_{S^n}^2+(\Delta_{S^n}f)^2\neq0$ and $\mbox{max}_{S^n}f/\mbox{min}_{S^n}f<\delta_n$, where $\delta_n=2^{1/n}$, when $n=2$ and $\delta_n=2^{1/(n-1)}$, when $n\ge3$. We then show that $f$ can be realized as the mean curvature of some conformal metric provided the Morse index counting condition holds for $f$. This shows that the possible best difference in the infinity norm may be the number $(\delta_n-1)/(\delta_n+1)$.