What's the defination of "quasi-conformal mappings"between Riemannian Manifolds?
Espectially I want to konw how the proof in section 4 of [Ha] works.
Thanks!


[Ha]R.S.Hamilton.Convex hypersurfaces with pinched second fundamental form.
Comm.Anal.Geom.,Vol2(1994),167-172.



What's the defination of "quasi-conformal mappings"between Riemannian Manifolds?
Here is the standard definition: Suppose that $M, N$ are oriented $n$-dimensional Riemannian manifolds, $f: M\to N$ is an orientation-preserving homeomorphism, which locally belongs to $W^{1,n}$. Then $f$ is called quasiconformal if there exists $K<\infty$ so that almost everywhere in $M$ the following inequality holds:
$$
||Df(x)||^n\le K |J\_f(x)|. 
$$
Here the left hand side is the operator norm of $Df(x): T\_xM\to T\_yN, y=f(x)$, and the Jacobian in the right hand-side is computed using isometric identification of $T\_xM, T\_yN$ with $R^n$. Depending on your taste, you can relax the orientation assumption. The number $K$ above is an upper bound on the *linear dilatation* of $f$. 


There are several other equivalent definitions, using conformal modulus or capacity, which are defined by repeating verbatim the Euclidean definitions. Things work analogously to the case of flat metrics, since, as metric spaces, Riemannian manifolds are *Loewner spaces.* You can find more on this, e.g. in Heinonen's book "Lectures on Analysis on Metric Spaces". 


