Let $U$ be a compact set in $\mathbb{R}^k$ and let $f:U\to\mathbb{R}^n$ be a $C^1$ bijection. Consider the manifold $M=f(U)$.(adsbygoogle = window.adsbygoogle || []).push({});

Its volume distortion is defined as $G=det(D_{f}^{t}D_{f}).$ If $n=1$, one can deduce that $G=1+|\nabla f|^2$.

What happens for $n>1$? Can one bound from below this $G$? If so: under which assumptions?

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# I Bounding the volume distortion of a manifold

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