Program of Courses

 

Giovanni Alberti: Introduction to minimal surfaces and finite perimeter sets.

In these lectures I will first recall the basic notions and results that are needed to study minimal surfaces in the smooth setting (above all the area formula and the first variation of the area), give a short review of the main (classical) techniques for existence results, and then outline the theory of Finite Perimeter Sets, including the main results of the theory (compactness, structure of distributional derivative, rectifiability). If time allows, I will conclude with a few applications.

                                                                                                                                                                                                                                                                            

 

Yoshihiro Tonegawa: Analysis on the mean curvature flow and the reaction-diffusion approximation

The course covers two separate but closely related topics. The first topic is the mean curvature flow in the framework of GMT due to Brakke. It is a flow of varifold moving by the generalized mean curvature. Starting from a quick review on the necessary tools and facts from GMT and the definition of the Brakke mean curvature flow, I will give an overview on the proof of the local regularity theorem. The second topic is the reaction-diffusion approximation of phase boundaries with key words such as the Modica-Mortola functional and the Allen-Cahn equation. Their singular perturbation problems are related to objects such as minimal surfaces and mean curvature flows in the framework of GMT.

Bibliography (Books):

Brakke, K., The Motion of a Surface by its Mean Curvature, Math. Notes 20, Princeton Univ. Press, Princeton, NJ, 1978

Ilmanen, T., Elliptic regularization and partial regularity for motion by mean curvature. Mem. Am. Math. Soc. 108(520) (1994)

Simon, L., Lectures on Geometric Measure Theory. Proc. Centre Math. Anal. Austral. Nat. Univ. 3 (1983)

Bibliography (Papers):

Hutchinson, J. E., Tonegawa, Y., Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory. Calc. Var. PDE 10(1), 49-84 (2000)

Ilmanen, T., Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature. J. Diff. Geom. 38(2), 417-461(1993)

Kasai, K., Tonegawa, Y., A general regularity theory for weak mean curvature flow, Calc. Var. PDE 50(1), 1-68 (2014)

                                                                                                                                                                                                                                                                            

 

Tatiana Toro: Geometry of measures and applications

In the 1920's Besicovitch studied linearly measurable sets in the plane, that is sets with locally finite "length". The basic question he addressed was whether the infinitesimal properties of the "length" of a set E in the plane yield geometric information on E itself. This simple question marks the beginning of the study of the geometry of measures and the associated field known as Geometric Measure Theory (GMT).

In this series of lectures we will present some of the main results in the area concerning the regularity of the support of a measure in terms of the behavior of its density or in terms of its tangent structure. We will discuss applications to PDEs, free boundary regularity problem and harmonic analysis. The aim is that the GMT component of the mini-course will be self contained.

References:

P. Mattila. Geometry of sets and measures in Euclidean spaces, Cambridge Stud. Adv. Math. 44, Cambridge Univ. Press, Cambridge, 1995.

D. Preiss, Geometry of measures in Rn: distribution, rectifiability, and densities, Ann. of Math. 125 (1987), 537–643.

On the smoothness of Hölder doubling measures, D. Preiss, X. Tolsa and T. Toro, Calculus of Variations and PDE's 35 (2009), 339-363.

Boundary Structure and size in terms of interior and exterior harmonic measures in higher dimensions with C. Kenig, D. Preiss and T. Toro, J. Amer. Math. Soc. 22 (2009), 771-796.

Regularity of Almost Minimizers with Free Boundary,G. David, and T. Toro to appear in Calculus of Variations and PDEs.

[AC] H. W. Alt & L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981), 105–144.

[ACF] H. W. Alt, L. A. Caffarelli & A. Friedman, Variational problems with two phases and their free boundaries, Trans. Amer. Math. Soc. 282 (1984), 431–461.

[CJK] L. A. Caffarelli, D. Jerison & C. Kenig, Global energy minimizers for free boundary problems and full regularity in three dimensions. Non-compact problems at the in- tersection of geometry, analysis, and topology, 8397, Contemp. Math., 350, Amer. Math. Soc., Providence, RI, 2004.

[DeJ] D. DeSilva & D. Jerison, A singular energy minimizing free boundary. J. Reine Angew. Math. 635 (2009), 121

                                                                                                                                                                                                                                                                            

 

Fernando Coda-Marques: Min-max theory and the solution to the Willmore conjecture

The Willmore conjecture, proposed in 1965, concerns the quest to find the best torus of all. It is posed as a global problem in the calculus of variations: that of minimizing the Willmore energy, defined as the total integral of the square of the mean curvature, of a surface of genus one in three-space. This is a conformally invariant variational problem that has inspired a lot of mathematics over the years, helping bringing together ideas from subjects like conformal geometry, partial differential equations, algebraic geometry and geometric measure theory.


In this minicourse we will present  our solution to the conjecture, obtained jointly with Andre Neves,   through the min-max approach. The key insight comes from the analysis of the geometric and topological properties of a new kind of sweepout:  a  five-dimensional family of surfaces in the three-sphere that detects the Clifford torus as a min-max minimal surface. The implementation of the program is based on the Almgren-Pitts min-max theory for the area functional, developed in
the framework  of Geometric Measure Theory.


Bibliography:

Colding, T., De Lellis, C., The min-max construction of minimal surfaces,  Surveys in  Differential Geometry VIII , International Press,  (2003),   75--107.

Marques, F. C.,  Neves A., Min-max theory and the Willmore conjecture, Ann. of Math. 179 2 (2014), 683--782.

Pitts, J., Existence and regularity of minimal surfaces on Riemannian manifolds, Mathematical Notes 27, Princeton University Press, Princeton, (1981). 

                                                                                                                                                                                                                                                        

 

Camillo De Lellis: Center manifolds and regularity of area-minimizing currents

 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

Γ={(z,w)∈C2 :(z−w2)2 =w2015}.

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 = w2}: 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 C3,α 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 C3,α regularity around regular points: in contrast the “classical proof” shows first C1,α 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 [1]. A much shorter derivation of the existence of a center manifold has been proposed recently by the author and Emanuele Spadaro in [3]. 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 [2].

References

[1] Frederick J. Almgren, Jr. Almgren’s big regularity paper, volume 1 of World Scientific Monograph Series in Mathematics. World Scientific Publishing Co. Inc., River Edge, NJ, 2000.

[2] Camillo De Lellis and Emanuele Spadaro. Center manifold: a case study. Discrete Contin. Dyn. Syst., 31(4):1249–1272, 2011.

[3] Camillo De Lellis and Emanuele Spadaro. Regularity of area-minimizing currents II: center manifold. Preprint, 2013. 1

 

                                                                                                                                                                                                                                                                            

 

Joseph Fu: Integral geometric regularity

In the original form given by Blaschke in the 1930s, the famous Principal Kinematic Formula expresses the Euler characteristic of the intersection of two sufficiently regular objects in euclidean space, integrated over the space of all possible relative positions, in terms of geometric invariants associated to each of them individually. It is natural to wonder about the precise regularity needed  for this to work. The question turns on the existence of the normal cycle  of such an object A, i.e. an integral current that stands in for its manifolds of unit normals if A is too irregular for the latter to exist in a literal sense. Despite significant recent progress, a comprehensive understanding of this construction remains maddeningly elusive. In these lectures we will discuss both of these aspects.

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