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Beyond the holomorphic maps such as the weistrass representation, is there a simple way to construct maps from $T^2$ to $S^2$ which is easy to deal with analyticly, say degree=1 and easy to calculate the differentials and integrals etc.?

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 Well, the degree can't be one because the two are not isomorphic as compact Riemann surfaces... – David Corwin Nov 22 at 2:57
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 Yes, very easy, of degree 0, for example, the constant map. – Alexandre Eremenko Nov 22 at 3:32
In degree zero one can do a lot better (meaning, generate more interesting maps, though I suppose the constant map has the virtue that it is not very hard to compute integrals): e.g., draw the torus in 3-space, and project it in some random way onto the plane. – Igor Rivin Nov 22 at 3:45
Yeah there are some similar ways such as the guass map of immersed torus. What I'm acutally interested is try to get some analytic estimates when the map is "nearly" holomorphic, i.e. its anti-holomorhpic energy is small. – zalver Nov 22 at 5:21
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 You should edit your question so that you ask what you really want, and include some more details (see for instance mathoverflow.net/howtoask ). Then it might possibly be reopened. – jc Nov 22 at 9:55

closed as not a real question by unknown (google), Will Sawin, Misha, Ryan Budney, Andres Caicedo Nov 22 at 6:14

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