Conformal mappings and isometric immersions under second order Sobolev regularity

TL;DRAbstract

We consider two classes of vector valued functions with conformal constraint- conformal mappings from an $n$-dimensional domain into $\bbbr^n$ and isometric immersions of an $n$-dimensional domain into $\bbbr^{n+1}$ (co-dimension one) for $n\geq 3$. Iwaniec and Martin proved that in even dimensions $n\geq 3$, $W_{\rm{loc}}^{1,n/2}$ conformal mappings are M\"{o}bius transformations and they conjectured that it should also be true in odd dimensions. In the first part of this manuscript, we prove this theorem for a conformal map $f\in W_{\rm{loc}}^{1,1}$ in dimension $n\geq 3$ under one additional assumption that the norm of the first order derivative $|Df|$ satisfies $|Df|^p\in W_{\rm{loc}}^{1,2}$ for $p\geq (n-2)/4$. This is optimal in the sense that if $|Df|^p\in W_{\rm{loc}}^{1,2}$ for $p< (n-2)/4$, it may not be a M\"{o}bius transform. This result shows the necessity of the Sobolev exponent in the Iwaniec-Martin conjecture. In the second part, we prove the developability and $C_{\rm

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