Please note special day (Tuesday), time and location.  The regularity of immersed submanifolds depends on the estimates on the extrinsic curvatures. In this talk, we discuss some new estimates for the second fundamental forms of embedded submanifolds in terms of intrinsic geometry. Some of these results are even new for the hypersurfaces (co-dimension one case). Key ingredients are drawn from recent works in the problem of prescribed curvature measures and regularity of curvature equations.