In contrast to classic minimal surface theory relatively little seems to be known about minimal surfaces in Finsler manifolds. We explore a connection between the Busemann-Hausdorff volume in Finsler spaces and Cartan functionals to prove new results in that direction, such as Bernstein theorems, a uniqueness result, and removability of singularities for Finsler-minimal graphs, isoperimetric inequalities and enclosure theorems for minimal immersions in Finsler space, and we treat the Plateau problem in Finsler 3-space.

We consider the 4-D Yang Mills heat flow and the 2-D harmonic map heat flow. In these dimensions, the associated energy functional is (locally) conformally invariant, that is, the dimension is critical. We discuss some similarities and differences between the two flows, in particular in the equivariant setting.