Maximum norm stability and error estimates for the evolving surface finite element method

We show convergence in the natural $L^{\infty}$- and $W^{1,\infty}$-norm for a semidiscretization with linear finite elements of a linear parabolic partial differential equations on evolving surfaces. To prove this we show error estimates for a Ritz map, error estimates for the material derivative of a Ritz map and a weak discrete maximum principle.