Thursday, 2 April 2009

cv.complex variables - conformally embedding complex tori into R^3

Let L be a lattice in mathbbC with two fundamental periods, so that mathbbC/L is topologically a torus. Let p:mathbbC/LmapstomathbbR3 be an embedding (C1, say). Call p conformal if pulling back the standard metric on mathbbR3 along p yields a metric in the equivalence class of metrics on mathbbC/L (i.e. a multiple of the identity matrix).




Is there an explicit formula for such a p in the case of L an oblique lattice?





Backgorund



The existence of such C1 embeddings is implied by the nash embedding theorem (fix a metric on mathbbC/L, pick any short embedding, apply nash iteration to make it isometric and hence conformal).



For orthogonal lattices, the solution is simple: Parametrise the standard torus of radii r1, r2 in the usual way. Make the ansatz pi(theta,phi)=(f(theta),h(phi)), pull back the standard metric on mathbbC/L and solve the resulting system of ODEs. This relates r1/r2 to the ratio of the magnitudes of the periods. This shows that no standard torus can be the image of p in the original question (oblique lattice), although that is geometrically clear anyway.

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