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I have been reading about harmonic map flow for maps from a Riemann surface. I presume a lot of the results are specific to 2D as the conformal invariance of the energy is crucial to the arguments. Would anyone be able to give a rough idea of which results have been generalised to higher dimensional domains?

For example:

(i) Is there a classification of a phenomena (or a selection of phenomena), analagous to the "bubbling" in 2D which must always be present for a singularity for the flow to form?

(ii) Since conformal harmonic maps in 2D give minimal surfaces, are there any results in higher dimensions which relate harmonic map flow to mean curvature flow? I understand that some conjectures for using mean curvature flow to prove the existence of special Lagrangians have been unanswered due to the tendancy of MCF to form singularities, and wonder if there is any hope to get around this by looking instead at harmonic map flow.

I wonder if in fact the correct integral to minimize in $n$ dimensions is not the energy, but rather something like $\int_M |\nabla u|^n $.

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