using a new set of ideas that combines dynamical systems techniques with control theory and viscosity solutions methods. In (1), H(p, x) : R → R is a smooth Hamiltonian, strictly convex, i.e., D vvL(x, v) > γ > 0 uniformly (this is also called uniformly convex by some authors), and coercive in p (lim|p|→∞ H(p,x) |p| = ∞), and Z n periodic in x (H(p, x + k) = H(p, x) for k ∈ Z). Since R is the u...

In this paper we investigate the rigidity of ancient solutions mean curvature flow with arbitrary codimension in space forms. We first prove that under certain sharp asymptotic pointwise pinching condition solution a sphere is either shrinking spherical cap or totally geodesic sphere. Then show hyperbolic family spheres. also obtain result for nonnegatively curved form an integral condition.