  In this paper, we first investigate a new locally constrained mean curvature
flow (1.5) and prove that if the initial hypersurface is of smoothly compact
starshaped, then the solution of the flow (1.5) exists for all time and
converges to a sphere in smooth topology. Following this flow argument, not
only do we achieve a new proof of the celebrated sharp Michael-Simon inequality
for mean curvature in (n+1) dimensional Euclidean space, but we also get the
necessary and sufficient condition for the establishment of the equality. In
the second part of this paper, we study a mean curvature type flow (1.7) of
static convex hypersurfaces in (n+1) dimensional Euclidean space, and prove
that the flow (1.7) has a unique smooth solution for all time t>0, and the
static convexity of the hypersurface is preserved along the flow (1.7).
Moreover, The solution of the flow (1.7) converges exponentially to a sphere of
radius R in smooth topology as time tends to infinity. By exploiting the
properties of this flow, we develop and present a new sharp Michael-Simon
inequality for kth mean curvature.
