A flat torus can be isometrically embedded in three-dimensional Euclidean space with a C1 map (by the Nash embedding theorem) but not with a C2 map, and the Clifford torus provides an isometric analytic embedding of a flat torus in four dimensions.
There is also an older method called Kantorovich iteration that uses Newton's method directly (without the introduction of smoothing operators).
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The basic idea of Nash's solution of the embedding problem is the use of Newton's method to prove the existence of a solution to the above system of PDEs.
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