Mu-Tao Wang (Math. Res. Lett. 2001) showed that any diffeomorphism $f:S^2\to S^2$ is isotopic to an isometry, which was originally shown by Smale (Proc. AMS 1959) 

Mao-Pei Tsui and Mu-Tao Wang (Comm. Pure Appl. Math. 2004) showed that if $f:S^n\to S^m$ is area-decreasing on two-dimensional submanifolds, then $f$ is null-homotopic. (Gromov had shown this in the weaker context that the two-dimensional area distortion factor is sufficiently close to 0. Larry Guth (Geom. Func. Anal. 2013) has counterexamples if "two" is replaced by "three".)

Ivana Medos and Mu-Tao Wang (J Diff. Geom. 2011) showed that if  $f:\mathbb{CP}^n\to\mathbb{CP}^n$ is a symplectomorphism such that $f$ and $f^{-1}$ have Lipschitz constants sufficiently close to one, then $f$ is symplectically isotopic to an isometry. (Gromov (Invent. Math. 1985) showed that in the case $n=2$ this is true without a condition on the Lipschitz factors.)

The method in each case is to deform the graph of $f$ by the mean curvature flow and to show a long-time existence and convergence result. So it is mean curvature in codimension larger than one, as opposed to most research in MCF.