Centre International de Mathématiques Pures et Appliquées



Ministère Enseignement Supérieur et Recherche, France UNESCO
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Minimal surfaces, overdeterminated problems and geometric analysis

English Version

CIMPA-ICTP-CHILE research school

Research School co-sponsored with ICTP

Others sponsors : ANR, PUC, CMM

PUC, Santiago, April 8-15, 2015

Report by Ahmad El Soufi

In this research school topics related to minimal surfaces, overdetermined problems and, more generally, geometric analysis will be studied. The main objective is to present to students and young researchers how tools from differential geometry and analysis of partial differential equations can be combined to obtain interesting, new results in both fields.

Minimal surfaces theory and geometric analysis are very active topics in Brazil. These theories are quite advanced and expect to spur developments in new areas. For example Allen-Cahn equation which models two phases transitions is a counterpart of the minimal surface equation in semilinear elliptic partial differential equations. Since the resolution of the De Gorgi conjecture by the group of non linear PDEs in Chile there have been growing interest in geometric aspects of semilinear elliptic equations. Regularity theory of a minimizer of an elliptic functional is at the origin of the subject, beginning with the work of Modica. De Giorgi’s conjecture is related to the Bernstein problem in minimal surface theory.

Other topics like overdetermined problems show a deep interaction between differential geometry, variational problems and PDE. Some classification of the space of Alexandrov embedded domain has been recently establish using Weierstrass representation and minimal surfaces techniques.

We expect to bring researchers and students in differential geometry from South America in view of promoting interaction with the Chilean PDE group.The activities of the school will include four mini-courses and several research talks. Although it is expected that participants should have working knowledge of basic aspects of differential geometry and analysis presentations will be accessible to students and, in general, to a public of non-experts in these topics. All expositors are leading experts on their subjects, which will give an opportunity to the participants to get acquainted with the basic techniques in the area and to be exposed to the current state of art.

Administrative and scientific coordinators

- Laurent Hauswirth (Université Paris-Est) hauswirth@univ-mlv.fr
- Mariel Saez (PUC, Chile) msaezt@mat.puc.cl

Scientific Committee

  • Camillo De Lellis (Zurich)
  • Manuel Del Pino (U. Chile)
  • Laurent Hauswirth (Paris-Est)
  • Michal Kowalczyk (U. Chile)
  • Rafe Mazzeo (Stanford)
  • Frank Pacard (Polytechnique)

Organizing Committee

  • Mariel Saez (PUC)
  • Michal Kowalczyk (U. Chile)

Scientific programme

Mini-courses :

Camillo De Lellis (4h) : "The theory of stationary varifolds"
Abstract : We will introduce stationary varifolds, a concept introduced first by Almgren and which has found quite deep applications in the last decades, most notably to the existence of critical points for the area functional. After reviewing some background in geometric measure theory, the main focus of the course will to prove Allard’s fundamental regularity theorem. I will mostly use the notes available at the web page

Frank Pacard (4h) : "Entire solutions of the Allen-Cahn equation in the plane."
Abstract : The Allen-Cahn equation appears in the modeling of a phase transition phenomena. In this series of lectures, I will describe the space of entire solutions of the Allen-Cahn equation which are defined the euclidean 2-plane. The solutions we are interested in have the property that, at infinity, their zero set is, away from a compact, asymptotic to a finite number of affine lines (which are called the « ends  » of the solutions). I will address the question of the moduli space theory for such solutions, the classification of 4-ended solutions and the construction of 2k- ended solutions using some gluing methods. All results and tools I will present are strongly influenced by results and tools in the theory of minimal and/or constant mean curvature surfaces in Euclidean 3-space.

Alberto Farina (3h) : "Splitting theorems and geometric aspects of overdeterminated problems"
Abstract : In this mini course I will present a unified approach to splitting theorems and to overdetermined elliptic boundary value problems in a non compact Riemannian setting.

Pieralberto Sicbaldi (2h) : "Geometry of overdetermined elliptic problems."
Abstract : Overdetermined elliptic systems appear in many problems in Physics and Applied Mathematics, and the classification of their solutions is a major topic in Analysis of PDEs. In the last years, a deep and surprising parallelism with minimal and constant mean curvature surfaces has been pointed out, and this suggests that the class of solutions to overdetermined elliptic problems is very interesting, and rich in geometric properties and structures. In these two lectures I will present the main tools to study overdetermined elliptic problems from the geometric point of view and I will try to underline the parallelism with minimal and constant mean curvature surfaces. In this last context, I will present many open problems, and reasonable potential results that could be obtained in the future.

Laurent Hauswirth (3h) : "Maximum principle on two points functions and application "
Abstract : In this lecture I will present recent techniques on two points function maximum principle introduced by Andrews and Clutterbuck in the proof of the fundamental gap property. I will discuss applications in geometry.

Conferences of Research :

B. Daniel (U. Lorraine) :
M. Chuaqui (PUC):Quasiconformal Extensions to Space of Weierstrass-Enneper Lifts
M. Del Pino (UChile) :
J. Espinar (Impa-Brazil) :
D. Henao (UC) :
A. Jimenez (UFRJ-Brazil):Isolated singularities of PDE´s and applications.
M. Kowalczyk (UChile)
J. Lira(Fortaleza-Brazil):Applications of the maximum principle to minimal graphs
A. Malchiodi:Embedded Willmore tori in three-manifolds with small area constraint
L. Mazet (Upec-France):Minimal hypersurfaces asymptotic to Simons cones.
M. Musso (PUC):Nondegeneracy of Nonradial Nodal Solutions to Yamabe Problem
J. Perez (Granada-Spain)):Existence of CMC foliations in compact n-manifolds
A. Quaas (USM) :
R. Rodiac (UPEC) :
H. Rosenberg (Impa-Brazil) :
M. Saez (PUC) :
D. Zhou (Niteroi-Brazil) :
T. Zolotareva (Polytechnique-France) :

Deadline for registration :

November 30, 2014

Application procedure only for applicants not from Chile.

Applicants from Chile must contact local organizer :
Mariel Saez (msaezt@mat.puc.cl)

Voir en ligne : Site web local de l’Ecole