22 jun 2015

8 hours course

Abstract.

A celebrated theorem of Almgren shows that every integer rectifiable current which minimizes (locally) the area is a smooth submanifold except for a singular set of codimension at most 2. Almgren’s theorem is sharp in codimension higher than 1, because holomorphic subvarieties of Cn are area-minimizing. In fact the typical singularity of a 2-dimensional area-minimizing current is modelled by branch points of holomorphic curves. These singularities are rather difficult to analyze because they might be very high order phenomena.

Take for instance the holomorphic curve \[ \Gamma=\{(z,w)\in \mathbb{C}^2 \, :\, (z−w^2)^2 =w^{2015}\}. \] The origin is a singular point but in any neighborhood of it Γ is extremely close to two copies of the smooth submanifold $\{(z,w)\, :\, z = w^2\}$: the “singular” behavior can be seen clearly only after we “mod out” such regular part.

The most involved part of Almgren’s proof is the construction, for a general area- minimizing current, of what he calls the “center manifold”: a $C^{3,\alpha}$ submanifold which approximates with a very high degree of precision the regular part of an area-minimizing current in a neighborhood of a singular point. A very interesting corollary of this construction is that we can conclude directly $C^{3,\alpha}$ regularity around regular points: in contrast the “classical proof” shows first $C^{1,\alpha}$ regularity and then boostraps using Schauder estimates.

The construction of the center manifold is undoubtedly the most difficult part of Almgren’s proof and it alone takes more than 500 pages in his original monograph. A much shorter derivation of the existence of a center manifold has been proposed recently by the author and Emanuele Spadaro. In this series of lectures I will highlight why one needs such an object and I will give some of the details of its construction in the simplified situation of regular points, following a self-contained account given in Discrete Contin. Dyn. Syst., 31(4):1249–1272, 2011.