Parametrized stationary varifolds and the multiplicity one conjecture

We present a new notion of "two-dimensional parametrized stationary varifold", which merges the classical parametrized pproach to the variational construction of minimal surfaces with the modern varifold approach. In contrast to the lack of regularity for general integer stationary varifolds, we will see how the subtle interaction between the parametrized structure and the corresponding natural stationarity condition allows to get an optimal regularity result, independent of the codimension and free of extra assumptions (such as stability). Such parametrized stationary varifolds arise as limiting objects produced by a penalization method to get immersed minimal surfaces out of min-maxes in the space of immersions. In this setting, we will see that the natural version of the multiplicity one conjecture by Marques and Neves always holds, without any genericity or codimension one assumption. From this we will deduce a general result for a min-max with k parameters, showing that it gives rise to a (branched, immersed) minimal surface with Morse index at most k. All of this is joint work with Tristan Rivière; the last statement relies also on previous work by T. Rivière (for the existence theory) and by A. Michelat (for the lower semicontinuity of the index).