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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!

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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.

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  • $\begingroup$ Thanks for your answer Sebastian. I understand that once you have a minimal surface in $T^3$, you get two spinors which define the holomoprhic Gauss map. But I don't think one can always recover the minimal surface from 2 spinors. It is proved in Meeks' paper "The theory of triply periodic minimal surfaces" that hyperelliptic surface of even genus cannot be minimally immersed into $T^3$. But certainly these surfaces admit spinor bundles with enough sections. And in the same paper, Meeks constructed a real 5-dimensional family of minimal embedded genus 2 surfaces in $T^3$. $\endgroup$
    – Piojo
    Commented Jul 1, 2015 at 14:48
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    $\begingroup$ By last line I mean genus 3. $\endgroup$
    – Piojo
    Commented Jul 1, 2015 at 15:40
  • $\begingroup$ Dear Piojo, I do not understand your comment. Whenever you have a minimal surface with translational periods only, you get two holomorphic spinors, and vice versa. The problem is to find two spinors such that the corresponding real parts of the 2g many $\mathbb C^3$ valued periods span a lattice in euclidean 3-space. $\endgroup$
    – Sebastian
    Commented Jul 1, 2015 at 19:52

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