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Let $S_1$ and $S_2$ be closed orientable surfaces with some metric defined on each such that the diameters and areas of $S_1$ and $S_2$ are bounded above by a constant $D$ and the injectivity radii of $S_1$ and $S_2$ are bounded below by a constant $\epsilon$.
Suppose $f:S_1 \to S_2$ is an $L$-Lipschitz map which is homotopic to an embedding. Can the lengths of the tracks of a homotopy from the immersion $f$ to an embedding be bounded above by a constant that depends only on $D$, $\epsilon$, and $L$?

Edit: The original question assumed that the original map was an immersion, but this implies (as pointed out in the comments) that the map is already an embedding.

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an immersion $f$ is a homeomorphism if and only if $\mathop{\rm deg}f=\pm 1$, and if $\mathop{\rm deg}f\neq\pm 1$ then $f$ is not homotopic to an embedding. Did I miss something? –  Anton Petrunin Oct 26 '11 at 4:43
    
Since $f$ is homotopic to an embedding, $\deg f = \pm 1$. If $H: S_1 \times [0,1] \to S_2$ is a homotopy from $f$ to an embedding, then for each $p\in S_1$ we get a path $\gamma_p (t) = H(p,t)$ in $S_2$. I want to know if there is a constant depending only on $D$, $\epsilon$, and $L$ that is an upper bound on the length of any such path. –  b b Oct 26 '11 at 6:27
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an immersion between closed surfaces is a covering and an embedding is a diffeomorphism. So if an immersion is homotopic to an embedding then it is a diffeomorphism already. The question becomes more interesting if you don't assume the original map to be an immersion. Then it seems to me that this should be true. your conditions imply bounds on diameters, genus and combinatorial complexity of surfaces involved. so I think that should mean that one can pass to equivalent hyperbolic surfaces (if genus is $>1$) and use simplex straightening to get a homotopy with controlled track lengths. –  Vitali Kapovitch Jan 5 '12 at 4:55
    
Thank you for the comments. I am indeed interested in the more general (and more interesting) case in which the original map is not necessarily an immersion. Passing to hyperbolic surfaces and using simplex straightening seems to do the trick for me. –  b b Jan 18 '12 at 4:15

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