Hamilton-Type Gradient Estimates for Porous Medium and Fast Diffusion Equations on Riemannian Manifolds for all exponents

We establish local Hamilton-type gradient estimates for positive $C^{2,1}$ solutions of $u_t=\mathrm{\Delta }{u}^m$ on complete Riemannian manifolds whose Ricci curvature is bounded from below. For every fixed $m>1$ and every fixed $0<m<1$, there exist $\beta =\beta (m,n)>0$ and $C=C(m,n)>0$ such that a solution $0<u⩽A$ in $B_{2R}(x_0)\times (t_0-T,t_0]$ satisfies the following local gradient estimate \begin{equation*} \sup_{B_R(x_0)\times(t_0-T/2,t_0]}|\nabla u^β| \leqslant C A^β\left(\frac1R+\sqrt{k}+\frac{A^{(1-m)/2}}{\sqrt T}\right), \end{equation*} where ${\mathrm{R}\mathrm{i}\mathrm{c}}_M⩾-k$. The proof uses an intrinsic quantitative alternative to locate, around each prescribed positive point, a cylinder on which the solution has a controlled upper-to-lower ratio. A local gradient estimate on this cylinder is combined with a stopping argument. As a consequence, every uniformly bounded positive ancient solution on a connected complete manifold with nonnegative Ricci curvature is constant.