On the Fill-in of Nonnegative scalar curvature metric

2019.12.26（星期四），14:00-15:00

点：数学院南楼N205

要：One basic problem in Riemannian geometry is to study : under what kind of conditions does the n-1 dim Bartnik data $(\Sigma, \gamma, H)$ admits a fill-in metric g with scalar curvature having a given lower bound ? That is, there is a compact Riemanina manifold $(\Omega^n, g)$ with boundary of scalar curvature $R_g\ge\sigma>-\infty$, and an isometry $\chi : (\Sigma^{n-1},\gamma)\to (\partial\Omega^n, g|_{\partial\Omega})$ so that $H=\chi^*H_{\partial\Omega}$, where $H_{\partial\Omega}$ is the mean curvature of $\partial\Omega$ in $(\Omega^n, g)$ with respect to the outward unit normal vector. In this talk, I will consider the problem of fill-in of nonnegative scalar curvature (NNSC) metric for a Bartnik data $(\Sigma, \gamma, H)$, and prove that given a metric $\gamma$ on $S^{n-1}\ (3\le n\le 7)$, $(S^{n-1}, \gamma, H)$ admits no fill-in with NNSC metric provided the prescribed mean curvature H is large enough; moreover, if $\gamma$ is positive scalar curvature (PSC) metric and is isotopic to the standard metric on $S^{n-1}$, the much weaker condition that the total mean curvature $\int_{S^{n-1}}Hd\mu$ is large enough rules out NNSC fill-in which give a partially affirmative answer to a conjecture due to Gromov. I will also talk about the relation between this problem and quasi-local mass in Math Relativity when n=3. The talk is based on recent joint work with Shi Yuguang, Wei Guodong and Zhu Jintian.