  We systematically investigate charge-ordering phases by means of a restricted
and unrestricted Gutzwiller approximation to the single-band Hubbard model with
nearest ($t$) and next-nearest neighbor hopping ($t'$). When $|t'/t|$ is small,
as appropriate for ${\rm La_{2-x}Sr_xCuO_4}$, stripes are found, whereas in
compounds with larger $|t'/t|$ (such as ${\rm Ca_{2-x}Na_x CuO_2Cl_2}$ and
${\rm Bi_2Sr_2CaCu_2O_{8+\delta}}$) checkerboard structures are favored. In
contrast to the linear doping dependence found for stripes the charge
periodicity of checkerboard textures is locked to 4 unit cells over a wide
doping range. In addition we find that checkerboard structures are favored at
surfaces.
