  We present a theory of optimal topological textures in nonlinear sigma-models
with degrees of freedom living in the Grassmannian $\mathrm{Gr}(M,N)$ manifold.
These textures describe skyrmion lattices of $N$-component fermions in a
quantising magnetic field, relevant to the physics of graphene, bilayer and
other multicomponent quantum Hall systems near integer filling factors $\nu>1$.
We derive analytically the optimality condition, minimizing topological charge
density fluctuations, for a general Grassmannian sigma model $\mathrm{Gr}(M,N)$
on a sphere and a torus, together with counting arguments which show that for
any filling factor and number of components there is a critical value of
topological charge $d_c$ above which there are no optimal textures. Below $d_c$
a solution of the optimality condition on a torus is unique, while in the case
of a sphere one has, in general, a continuum of solutions corresponding to new
{\it non-Goldstone} zero modes, whose degeneracy is not lifted (via a order
from disorder mechanism) by any fermion interactions depending only on the
distance on a sphere. We supplement our general theoretical considerations with
the exact analytical results for the case of $\mathrm{Gr}(2,4)$, appropriate
for recent experiments in graphene.
