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I think the conformal mapping theory in the plane are quit interesting and useful in physics. I learned that there is very few conformal mappings in higher dimensions, is there any reason for that?

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Usually in mathematics, if an object enjoys too many good properties, then there will be some kind of rigidity result for it.

Conformal mappings are perfect in two dimension and they are solutions to the so-called Beltrami system, which is highly over-determined when n>3. Thus, it is very rigid. By a result of Liouville, conformal mappings are restrictions of Mobius transformations in dimenions higher than two. For a proof of this result, I recommend the excellent book by Iwaniec and Martin,

Iwaniec, Tadeusz; Martin, Gaven Geometric function theory and non-linear analysis. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2001.

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