Analysis Aspects of Minimal Surfaces Spring 2020
The talks are now recorded. The videos are available here.
Some notes written by the speakers are available in the schedule table below (providing notes is not compulsory).
Organizers
Prof. Dr. Tristan Rivière, Alessandro Pigati
Overview
The goal of the seminar is to present the (by now) classical regularity theory for area minimizing codimension one minimal surfaces.
In the course of the seminar we will go through fundamental notions from geometric measure theory (rectifiability, coarea formula, finite perimeter sets, currents, varifolds, ...), as well as classical notions from elliptic regularity theory (excess decay, Caccioppoli/reverse Poincaré inequality, harmonic approximations, tangent cones, Federer's dimension reduction argument, ...).
One of the goals will be to have a completely explained proof of the smoothness of perimeter minimizing sets up to ambient dimension 7, and to explain why dimension 8 is a bifurcation point in the regularity theory.
Prerequisites
A good knowledge of Sobolev functions in several variables (as provided, e.g., by Functional Analysis I/II) is recommended.
Basic notions from differential geometry will be also assumed.
Registration
Due to the limited number of places, in order to participate to this reading seminar you have to register on
myStudies.
The deadline for the registration is February 13.
Where and when
The talks will take place every
Thursday, from 13.15 to 15.00, in the
Hermann Weyl Zimmer (HG G 43).
Schedule of the talks
The following is a provisional schedule of the talks. The first few ones will be given by the organizers, then the participants will speak (a couple every week).
| Week number | Date | Topic | Speaker | Notes |
|---|
| 1 | February 20 | Rectifiable sets | Tristan Rivière | .pdf |
| 2 | February 27 | Sets of finite perimeter | Yujie Wu | .pdf |
| 3 | March 12 | De Giorgi's structure theorem | Riccardo Caniato | .pdf |
| 4 | March 19 | First variation of area and monotonicity. Regularity of almost flat stationary sets: tilt-excess inequality | Alessandro Pigati | .pdf [last updated April 23] |
| 5 | March 26 | Regularity of almost flat stationary sets: Lipschitz and harmonic approximation | Alessandro Pigati | (same file) |
| 6 | April 2 | Regularity of almost flat stationary sets: conclusion. Proof of monotonicity formula | Alessandro Pigati | (same file) |
| 7 | April 9 | Currents and varifolds: first variation, monotonicity and its consequences. Compactness of rectifiable and integral varifolds
[slides] | Anthony Salib | |
| 8 | April 23 | Regularity of "multiplicity one" varifolds with small excess
[video] | David Morselli | .pdf |
| 9 | April 30 | Generalized varifolds and a rectifiability criterion by Ambrosio-Soner | Matteo Giardi | |
| 10 | May 7 | Minimality of the Simons cone | Matilde Gianocca | |
| 11 | May 14 | Improved regularity for area-minimizing currents in codimension one (a.k.a. perimeter-minimizing sets), part I | Seyed Ali Naseri Sadr | |
| 12 | May 28 | Improved regularity for area-minimizing currents in codimension one, part II | Alireza Ataei |
If you see your name misspelled, or assigned to the wrong week, please send an email to Alessandro Pigati.
Literature