  This work is concerned with the equation $ \partial_t \rho = \Delta_x \rho^m
$, $ m > 1 $, known as the porous medium equation. It shows stability of the
pressure of solutions close to flat travelling wave fronts in the homogeneous
Lipschitz sense that is in a way optimal for the treatment of the equation.
This is the first result of this type and implies global regularity estimates
for any number of derivatives of the pressure. Consequences include smoothness,
analyticity in temporal and tangential directions, and analyticity of the
interface between empty and occupied regions. In the course of the argument a
Gaussian estimate in an intrinsically arising space of homogeneous type is
crucial to obtain linear estimates by means of the non-Euclidean
Calder\'{o}n-Zygmund singular integral theory.
