Abstract：Let (\mathrm{\Sigma}^{n-1},\gamma) be an orientable  (n-1)-dimensional Riemannian manifold,  H  be a positive function on \mathrm{\Sigma}^{n-1}, it is  natural to ask: under what  conditions is it that  \gamma is  induced  by a Riemannian metric  g  with  nonnegative  scalar curvature, for example, defined on \mathrm{\Omega}^n, and   H is the mean curvature of \mathrm{\Sigma} in (\mathrm{\Omega}^n,g) with respect to the outward unit normal vector?  Recently, M.Gromov proposed several  conjectures on this question. In this first part of this talk I will describe what the conjectures are and survey some known results in this direction when n=3; In the second part of the talk,  I will present my several  recent results on this which  joint with Wang Wenlong, Wei guodong ， and Zhu jintian. I will also propose some problems relate to Gromov’s  conjectures.