Let $f: M \to N$ be a diffeomorphism between two riemannian Manifolds. Suppose there exist constants $0 < c \leq C$ such that for all $p \in M$, we have $c \leq |df\_p| \leq C$. Here, $df$ denotes the differential of $f$ as usual and $|df\_p|$ denotes the energy density of $f$ at $p$, i.e, the norm coming from the induced riemannian metric on the bundle $TM \otimes f^\*(TN)$. I wonder now if it's true that there are constants $0 < b \leq B$ with $b \leq |df^{-1}\_q| \leq B$ for every $q \in N$. 



Estimate for differential of inverse map
Lower bound yes. Upper bound no. 


If you take orthonormal frames for $M$ and $N$, then the norm $|df\_p|$ is the Frobenius norm on the matrix expression in those frames. Note that in any bases $df\_p$ and $df^{-1}\_{f(p)}$ are inverse matrices. The upper bound on the Frobenius norm of $df\_p$ implies that each of the singular values are bounded above by $C$, and hence their inverses are bounded below by $1/C$, which gives the lower bound. 


To see that the upper bound need not hold, let $M = N = \mathbb{R}^2$ and take


$$ (u,w) = f(x,y) = (x, \sinh^{-1}(y) ) $$


For $|df|$ we have the bound $ 1 \leq |df| \leq \sqrt{2}$. 


For $|df^{-1}|$ you can compute


$$ df^{-1}(u,w) = \begin{pmatrix} 1 & 0 \\ 0 & \cosh w \end{pmatrix} $$


is not bounded. 




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You can of course assume that $M, N$ are compact, but then the desired results are entirely trivial. 


