A Pogorelov estimate and a Liouville type theorem to parabolic $k$-Hessian equations

We consider Pogorelov type estimates and Liouville type theorems to parabolic $k$-Hessian equations of the form $-u_t{\sigma }_k(D^{2}u)=1$ in ${\mathbb{R}}^n\times (-\mathrm{\infty },0]$. We derive that any $k+1\mathbf{\text{-convex-monotone}}$ solution to $-u_t{\sigma }_k(D^{2}u)=1$ when $u(x,0)$ satisfies a quadratic growth and $0<m_1\le -u_t\le m_2$ must be a linear function of $t$ plus a quadratic polynomial of $x$.