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.
As for mean curvature flow in codimension one, any smooth compact four-manifold which is homeomorphic to $S^4$ can be smoothly embedded in $\mathbb{R}^5$, and I think some people hope that a sufficiently good understanding of its mean curvature flow could prove the four-dimensional smooth Poincaré conjecture, roughly analogously to the Hamilton-Perelman proof of the three-dimensional Poincaré conjecture using Ricci flow. But it would probably be much more complicated, for analytic reasons.