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We study interior $C^{1, \al}$ regularity of viscosity solutions of theparabolic Monge-Amp\'ere equation $$u_t = b(x,t) \ddua,$$ with exponent $p >0$and with coefficients $b$ which are bounded and measurable. We show that when$p$ is less than the critical power $\frac{1}{n-2}$ then solutions becomeinstantly $C^{1, \al}$ in the interior. Also, we prove the same result for anypower $p>0$ at those points where either the solution separates from theinitial data, or where the initial data is $C^{1, \beta}$.

$C^{1,\al}$ regularity of solutions to parabolic Monge-Amp\'ere equations