New min-max frameworks for minimal submanifolds in dimension two or codimension two

In this dissertation we develop new variational methods with the aim to build minimal k-submanifolds S in a closed ambient Riemannian manifold (M,g), by means of min-max procedures. We will focus almost exclusively on the cases k=2 and k=m-2, namely minimal surfaces and minimal submanifolds of codimension 2. After a general introduction, in the second chapter we present a recent min-max theory devised by the supervisor of this thesis, T. Rivière: starting from immersions which are critical for a suitable relaxation of the area, involving a power of the second fundamental form, this method builds, as the viscosity parameter tends to zero, a limit object satisfying a certain weak notion of minimality. The method applies to any min-max problem for the area functional in the space of immersions. We revisit this theory and show how to adapt it to the free boundary case, relative to any submanifold N of M; the outcome is essentially an immersed manifold which is critical with respect to the constraint that the boundary is contained in N. In the following chapter we study axiomatically this new weak notion of minimal immersion, whose instances are called parametrized stationary varifolds. These special varifolds are induced by a parametrization and a Borel multiplicity; they enjoy a localization property for the stationarity, with respect to the domain. In spite of the lack of general regularity results for stationary varifolds, this last property can be exploited to show that the multiplicity is constant and the parametrization is smooth, without any assumption on the codimension. The attention then moves to the study of the multiplicity when the parametrized varifold arises as a limit of critical or almost critical immersions for the relaxed functionals; in this case we show that the multiplicity is always equal to one. This result allows to obtain upper bounds on the Morse index of the limit minimal immersion, namely a bound on the instability of this map in the space of immersions. This fact would allow, by itself, to simplify the regularity theory, but its proof relies strongly on the theory contained in the previous chapters. In the last chapter, which is completely independent of the rest of the thesis, we present another theory meant to produce minimal submanifolds in codimension 2. It is inspired by the recent use of the Allen-Cahn functional for real-valued maps u on M, viewed as a relaxation of the (m-1)-area of the level sets of u: Rather than using the most immediate generalization given by the Ginzburg-Landau energy for complex-valued maps u, which makes the asymptotic analysis difficult and not completely satisfactory, we study instead the Yang-Mills-Higgs functional for couples (u,D), where u is a section of a given Hermitian line bundle L and D is a connection on it. This energy enjoys a natural gauge invariance for the symmetry group U(1). The study of this functional brings to a simpler analysis of the energy concentration set and, differently from what happens with Ginzburg-Landau, it allows to obtain the integrality and concentration of the limit varifold. The results contained in the third and fourth chapters have been obtained in collaboration with T. Rivière, while the ones from the last chapter come from a collaboration with D. Stern