题目：   Some remarks on the symmetry  of complete, locally conformally flat metrics on canonical domains of the round sphere with constant $Q$-curvature

报告人：  Zheng-Chao Han      (Rutgers Univ.)

摘要：We will report on some results, jointly with Alice Chang and Paul Yang of Princeton University, on the symmetry  of complete, locally conformally flat metrics on canonical domains of the round sphere with constant $Q$-curvature. More specifically

\begin{theorem}

Any complete, conformal metric $g$ on $\mathbb S^n \setminus \mathbb S^{l}$ for $l\le \frac{n-2}{2}$ satisfying

\begin{equation}\label{q1}

Q_g \equiv 1\; \text{or \ $0$},

\end{equation}

and

\begin{equation}\label{sp}

R_g \ge 0,

\end{equation}

in $\mathbb S^n \setminus \mathbb S^{l}$ has to be symmetric with respect to rotations of $\mathbb S^n$ which leave $\mathbb S^l$ invariant.

\end{theorem}

This theorem is a corollary of the following

\begin{theorem}\label{thm1}

Let $g$ be a conformal, complete metric on $\Omega \subsetneqq \mathbb S^n$ such that \eqref{q1} and \eqref{sp} hold in $\Omega$. Then for any ball $B\subset \subset \Omega$ in the canonical metric $g_{\mathbb S^n}$, the mean curvature of its boundary $\partial B$ in metric $g$ with respect to its inner normal is nonnegative.

\end{theorem}

时间： 2018年4月4日 14:00--16:00

地点：蒙民伟楼1105室

邀请人：陈学长 老师