  Maximal parabolic $L^p$-regularity of linear parabolic equations on an
evolving surface is shown by pulling back the problem to the initial surface
and studying the maximal $L^p$-regularity on a fixed surface. By freezing the
coefficients in the parabolic equations at a fixed time and utilizing a
perturbation argument around the freezed time, it is shown that backward
difference time discretizations of linear parabolic equations on an evolving
surface along characteristic trajectories can preserve maximal $L^p$-regularity
in the discrete setting. The result is applied to prove the stability and
convergence of time discretizations of nonlinear parabolic equations on an
evolving surface, with linearly implicit backward differentiation formulae
characteristic trajectories of the surface, for general locally Lipschitz
nonlinearities. The discrete maximal $L^p$-regularity is used to prove the
boundedness and stability of numerical solutions in the
$L^\infty(0,T;W^{1,\infty})$ norm, which is used to bound the nonlinear terms
in the stability analysis. Optimal-order error estimates of time
discretizations in the $L^\infty(0,T;W^{1,\infty})$ norm is obtained by
combining the stability analysis with the consistency estimates.
