I will give an introduction to the Teichmueller harmonic map flow, which is related both to the harmonic map flow and the mean curvature flow. I will describe some work, completed earlier this year, establishing that the large time effect of the flow is to decompose the flow map into branched minimal immersions with no loss of energy. Joint work with Tobias Huxol, Melanie Rupflin and Miaomiao Zhu.

In this talk I will discuss some rigidity results and obstructions on the topology at infinity of translating solitons of the mean curvature flow in the Euclidean space. These results were recently obtained in collaboration with M. Rimoldi and our approach relies on the theory of f-minimal hypersurfaces.