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.
