Can every hyperelliptic genus 3 surface be minimally immersed in flat $T^3$ Every minimally immersed genus 3 surface in flat $T^3$ must be hyperelliptic, as the Gauss map gives the degree 2 covering map. How about the converse of this problem? 
The only thing I can find is that in Meeks's paper he constructed a real 5-dimensional family of minimal genus 3 surface in $T^3$.
Thanks!
 A: This is not a full answer, but only a very naive dimension count:
minimally immersed (non-planar) surfaces into flat  $T^3$ are given by a pair of (linear independent) holomorphic spinors (in the same holomorphic spin bundle $S$ satisfying $S^2=K$). For a genus $3$ surface, every spin bundle has degree $2$ and has at most a two dimensional space of holomorphic sections. 
Moreover, the space of spin bundles is discrete (there exists exactly $2^8$ spin bundles on a genus 3 surface).
The complex 2 dimensional space of sections gives rise to a real 4 dimensional space of (locally defined and geometrically distinct) minimal surfaces: $GL(2,\mathbb C)$ acts on the 2d space, but the $SU(2)$ action gives rise to rotation of the minimal surface in euclidean 3space. Moreover, a mutual scaling of the spinors only rescales the minimal surface in euclidean 3space.
Hence, there are only "4 dimensions left".
The space of hyper-elliptic surfaces of genus 3 is complex 5 dimensional, and altogether we obtain a 14 dimensional space of minimal surfaces of genus 3 with periods. The conditions that the six $\mathbb R^3$-valued periods span
a lattice in $\mathbb R^3$ seem to be 9 real conditions. In order to answer your question in the negative, you should show that for these 9 real conditions there are at least 5 independent ones. If all conditions are independent (generically) you would get a real 5 dimensional space.
