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Sebastian
Sebastian
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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.

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, 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.

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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Sebastian
Sebastian
  • 6.8k
  • 1
  • 26
  • 32

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, 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.