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In this paper we study the classification of ancient convex solutions to the mean curvature flow in Rn+1 . An open problem related to the classification of type II singularities is whether a convex translating solution is k-rotationally symmetric for some integer 2 ≤ k ≤ n, namely whether its level set is a sphere or cylinder S k−1 × R n−k . In this paper we give an affirmative answer for entire solutions in dimension 2. In high dimensions we prove that there exist nonrotationally symmetric, entire convex translating solutions, but the blow-down in space of any entire convex translating solution is k-rotationally symmetric. We also prove that the blow-down in space-time of an ancient convex solution which sweeps the whole space R n+1 is a shrinking sphere or cylinder.

By studying a negative gradient flow of certain Hessian functionals we establish the existence of critical points of the functionals and consequently the existence of ground states to a class of nonhomogenous Hessian equations. To achieve this we derive uniform, first-and second-order a priori estimates for the elliptic and parabolic Hessian equations. Our results generalize well-known results for semilinear elliptic equations and the Monge-Ampère equation.