  In this paper we first obtain a constant rank theorem for the second
fundamental form of the space-time level sets of a space-time quasiconcave
solution of the heat equation. Utilizing this constant rank theorem, we can
obtain some strictly convexity results of the spatial and space-time level sets
of the space-time quasiconcave solution of the heat equation in a convex ring.
To explain our ideas and for completeness, we also review the constant rank
theorem technique for the space-time Hessian of space-time convex solution of
heat equation and for the second fundamental form of the convex level sets for
harmonic function.
