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(EDIT 1: Replaced hand-waving argument in third paragraph with a hopefully less incorrect version)

(EDIT 2: Added final paragraph about obtaining all conformal deformations for surfaces other than sphere.)

What I think you can also try and see if for other embedded surfaces, all infinitesimal deformations of conformal structure are accounted for by deformations of the embedding in a similar way.

For the case of S2, what you want is to do is take a normal vector field V (i.e. infinitesimal change of embedding) and produce a tangent vector field X such that flowing along X gives the same infinitesimal change in conformal structure as flowing along V. This should amount to solving a linear PDE, so as Dmitri says a PDE is definitely involved, but probably not as hard as proving the existence of isothermal coordinates (which from memory is non-linear). For the standard embedding of S2 there can't be too many choices for this linear differential operator given that it has to respect the SO(3)-symmetry.

I guess we're looking for a first-order equivariant linear operator from normal vector fields to tangent vector fields. If we identify normal fields with functions then two possible candidates are to take X=grad V or X to be the Hamiltonian flow generated by V. I can't think of any others and probably it's possible to prove these are the only such ones. (Assuming it's elliptic, the symbol of the operator must be an SO(3)-equivariant isomorphism from T*S2 to TS2 and there can't be too many choices! Using the metric leads to grad and using the area form leads to the Hamiltonian flow.) Then you just have to decide which one to use.

For the case of a general embedded surface $M$, you can ask "is it possible to obtain all deformations of conformal structure by deforming the embedding into R3?" To answer this we can again think of a normal vector field as a function V on the surface. There is a second-order linear differential operator D\colon C^\infty(M) \to \Omega^{0,1}(T)$$ which sends a normal vector field to the corresponding infinitesimal change of conformal structure.

(EDIT: Replaced hand-waving argument in last paragraph This operator will factor through the hessian with a hopefully less incorrect version)homomorphism from $T^* \otimes T^*$ to $T^{*0,1}\otimes T^{1,0}$. The operator $D$ will not be onto, but what we want to know is whether every cohomology class in $H^{0,1}(T)$ has a representative in the image of $D$. At least, this is how I would try and approach the question; I'm sure there are other methods.
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Perhaps

I think it might be is possible to see the infinitesimal rigidity of the spheresomehow, even if it does involve a PDE as Dmitri says.

Given that, it's tempting

I guess we're looking for a first-order equivariant linear operator from normal vector fields to tangent vector fields. If we identify normal fields with functions then two possible candidates are to take X=grad V where we or X to be the Hamiltonian flow generated by V. I can't think of V as a function on the sphere times the unit normal any others and then take its gradient with respect probably it's possible to the round metric on prove these are the sphereonly such ones. I've no idea if that's right(Assuming it's elliptic, however. I guess that flowing along grad V gives an infinitesimal change in the conformal structure symbol of $\Delta V$ (where we use the Laplacian of operator must be an SO(3)-equivariant isomorphism from T*S2 to TS2 and there can't be too many choices! Using the round metric on the sphere here). This also at least looks the right order for leads to grad and using the infinitesimal change in area form leads to the metric given by flowing along V, but I don't see immediately that these guesses are correct. At this point I would Hamiltonian flow.) Then you just have to compute things to see if it's right but perhaps I'll leave that to you! decide which it is.

(EDIT: Replaced hand-waving argument in last paragraph with a hopefully less incorrect version)
show/hide this revision's text 2 minor rewording

Perhaps it might be possible to see the infinitesimal rigidity of the sphere somehow, even if it does involve a PDE as Dmitri says.

What you want is to do is take a normal vector field V (i.e. infinitesimal change of embedding) and produce a tangent vector field X such that flowing along X gives the same infinitesimal change in conformal structure as flowing along V. This should amount to solving a linear PDE, so as Dmitri says a PDE is definitely involved, but probably not as hard as proving the existence of isothermal coordinates (which from memory is non-linear). For the standard embedding of S2 there can't be too many choices for this linear differential operator given that it has to respect the SO(3)-symmetry.

Given that, it's tempting to take X=grad V where we think of V as a function on the sphere times the unit normal and then take its gradient with respect to the round metric on the sphere. I've no idea if that's right, however. Flowing I guess that flowing along grad V gives an infinitesimal change in the conformal structure of $\Delta V$ (where we use the Laplacian of the round metric on the sphere here). This also at least looks the right order for the infinitesimal change in the metric given by flowing along V, but I don't see immediately that this is these guesses are correct. At this point I would have to compute the change things to see if it's right but perhaps I'll leave that to you!

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