Homogenization of Maxwell equations on manifolds of complicated microstructure

The Cauchy problem for the homogeneous system of the Maxwell equations on four-dimensional manifolds $\tilde M_\varepsilon^4=R_+^1\times M_\varepsilon^3$, where $M_\varepsilon^3$ are Riemannian manifolds of a complicated microstructure is considered. $M_\varepsilon^3$ consist of several copies of the space $R^3$ with a large number of holes attached by means of thin tubes. The dependence on a small parameter $\varepsilon>0$ is such that the number of tubes increases and their thickness vanishes, as $\varepsilon\to 0$. The asymptotic behaviour of electromagnetic field without charges and currents on $\tilde M_\varepsilon^4$ is studied as $\varepsilon\to 0$, and it is obtained that the density of electric charge appears in the Maxwell equations as a result of homogenization.