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edited Mar 10, 2017 at 9:42

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Since my previous answer to this question major progress was made: As Robert Haslhofer already mentioned, Brendle proved the Lawson conjecture that the only minimal CMC torus in the 3-sphere is the Clifford torus. Building on this work, Andrews and Li ("Embedded constant mean curvature tori in the three-sphere." J. Differential Geom. 99 (2015)) classified all embedded CMC tori in $S^3.$ To be more concrete, they used the same two point function as in Brendles proof in order to show that every embedded CMC torus must be rotationally symmetric, and hence is classified and can be written down in terms of elliptic functions explicitly.

The case of higher genus CMC surfaces in space forms is more difficult and unsolved by now. But there has been done computer experiments which suggest that the space of embedded CMC surfaces of higher genus $g\geq2$ with certain symmetries is related to the space of embedded CMC tori, see http://arxiv.org/pdf/1503.07838.pdf and the following image:

http://www.math.uni-tuebingen.de/ab/GeometrieWerkstatt/moduli.png $enter image description here$

Since my previous answer to this question major progress was made: As Robert Haslhofer already mentioned, Brendle proved the Lawson conjecture that the only minimal CMC torus in the 3-sphere is the Clifford torus. Building on this work, Andrews and Li ("Embedded constant mean curvature tori in the three-sphere." J. Differential Geom. 99 (2015)) classified all embedded CMC tori in $S^3.$ To be more concrete, they used the same two point function as in Brendles proof in order to show that every embedded CMC torus must be rotationally symmetric, and hence is classified and can be written down in terms of elliptic functions explicitly.

The case of higher genus CMC surfaces in space forms is more difficult and unsolved by now. But there has been done computer experiments which suggest that the space of embedded CMC surfaces of higher genus $g\geq2$ with certain symmetries is related to the space of embedded CMC tori, see http://arxiv.org/pdf/1503.07838.pdf and the following image:

http://www.math.uni-tuebingen.de/ab/GeometrieWerkstatt/moduli.png

Since my previous answer to this question major progress was made: As Robert Haslhofer already mentioned, Brendle proved the Lawson conjecture that the only minimal CMC torus in the 3-sphere is the Clifford torus. Building on this work, Andrews and Li ("Embedded constant mean curvature tori in the three-sphere." J. Differential Geom. 99 (2015)) classified all embedded CMC tori in $S^3.$ To be more concrete, they used the same two point function as in Brendles proof in order to show that every embedded CMC torus must be rotationally symmetric, and hence is classified and can be written down in terms of elliptic functions explicitly.

The case of higher genus CMC surfaces in space forms is more difficult and unsolved by now. But there has been done computer experiments which suggest that the space of embedded CMC surfaces of higher genus $g\geq2$ with certain symmetries is related to the space of embedded CMC tori, see http://arxiv.org/pdf/1503.07838.pdf and the following image:

$enter image description here$

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created Jan 21, 2016 at 11:45

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Since my previous answer to this question major progress was made: As Robert Haslhofer already mentioned, Brendle proved the Lawson conjecture that the only minimal CMC torus in the 3-sphere is the Clifford torus. Building on this work, Andrews and Li ("Embedded constant mean curvature tori in the three-sphere." J. Differential Geom. 99 (2015)) classified all embedded CMC tori in $S^3.$ To be more concrete, they used the same two point function as in Brendles proof in order to show that every embedded CMC torus must be rotationally symmetric, and hence is classified and can be written down in terms of elliptic functions explicitly.

The case of higher genus CMC surfaces in space forms is more difficult and unsolved by now. But there has been done computer experiments which suggest that the space of embedded CMC surfaces of higher genus $g\geq2$ with certain symmetries is related to the space of embedded CMC tori, see http://arxiv.org/pdf/1503.07838.pdf and the following image:

http://www.math.uni-tuebingen.de/ab/GeometrieWerkstatt/moduli.png