For n≥3, any short immersion of an n-dimensional domain into R^{2n} can be uniformly approximated by C^{1,θ} isometric immersions with θ close to 1/n in odd dimensions and 1/(n+1) in even dimensions.
Borisov,C 1,α-isometric immersions of Riemannian spaces
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A Nash-Kuiper theorem for isometric immersions in a high codimension
For n≥3, any short immersion of an n-dimensional domain into R^{2n} can be uniformly approximated by C^{1,θ} isometric immersions with θ close to 1/n in odd dimensions and 1/(n+1) in even dimensions.