ICTPCIMPA RESEARCH SCHOOL IN SANTIAGO DEL CHILE 

MINIMAL SURFACES, OVERDETERMINATED PROBLEMS AND GEOMETRIC ANALYSIS

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 AllenCahn 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 minicourses 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 nonexperts 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. Scientific committee:Local organizing comittee:Administrative and Scientific coordinator:Minicourses :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 http://www.math.uzh.ch/fileadmin/user/delellis/publikation/allard_31.pdfAbstract: The AllenCahn 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 AllenCahn equation which are defined the euclidean 2plane. 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 4ended 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 3space. 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. AbstractFarina ConferenceFarinaAbstract: 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. Abstract: In this lecture I will present recent techniques on two points function maximum principle. We will discuss estimate from below of the first eigenfunction of the Laplacian on manifolds with positive Ricci curvature in function of the diameter and we discuss how maximum principle techniques on two variables functions can improve this estimate. We will explain how B. Andrews and J. Clutterbuck use this technique to find the optimal minoration of the gap between the two first eigenvalues of a convex domain of the Euclidean space. In the remaining lecture I will describe applications in geometry: the short proof of B. Andrews for the nocollapsing property of mean curvature flow for mean convex Alexandrov embedded surfaces and the Lawson conjecture proved by S. Brendle. This lecture is inspired by a recent Bourbaki seminar written by G. Carron. Lectures and conferences will be held at PUC in Santiago del Chile Participants:list of participantsConferences of Research:The Conference will finish a priori wesdnesday 15, april at 13H00. program april 11 at CMM (U. Chile) program april 13 at CMM (U. Chile) B. Daniel (U. Lorraine): On the area of minimal surfaces in a slab M. Chuaqui (PUC):Quasiconformal Extensions to Space of WeierstrassEnneper Lifts M. Del Pino (UChile):Bubbling in the critical heat equation: the role of Green's function J. Espinar (ImpaBrazil):Geometry and topology of fextremal domains in Hadamard Manifold D. Henao (UC):Phasefield regularization of fracture surfaces in elasticity A. Jimenez (UFRJBrazil):Isolated singularities of PDE´s and applications. M. Kowalczyk (UChile):Delaunay ends solutions to the CahnHilliard equation J. Lira(FortalezaBrazil):Applications of the maximum principle to minimal graphs A. Malchiodi:Embedded Willmore tori in threemanifolds with small area constraint L. Mazet (UpecFrance):Minimal hypersurfaces asymptotic to Simons cones. M. Musso (PUC):Nondegeneracy of Nonradial Nodal Solutions to Yamabe Problem J. Perez (GranadaSpain)):Existence of CMC foliations in compact nmanifolds A. Quaas (USM):Liouville type theorems for nonlinear elliptic equations and systems involving fractional Laplacian in the halfspace R. Rodiac (UPEC):Harmonic maps with prescribed degrees on the boundary of an annulus and bifurcation of catenoid H. Rosenberg (ImpaBrazil): Minimal surfaces of least area and applications M. Saez (PUC):Mean curvature flow without singularities D. Zhou (NiteroiBrazil):Spectrum of the drifted Laplacian and applications T. Zolotareva (PolytechniqueFrance):Free Boundary Minimal Surfaces in the Unit 3Ball Lectures will be held at PUC and CMM in Santiago del Chile Practical informations on Lodging and Arrival:The hotel can be reached from the airport by taxi, shuttle or bus. The taxi and the shuttle can be hired after customs, before exiting the security area or outside. If you intend to pay with credit card you should hire it before exiting.The costs are 7000 pesos for the shuttle 20000 pesos for a taxi Further information can be found at the following links http://www.aeropuertosantiago.cl/english/mediosdetransportedesdeyhaciascl/taxis.html http://www.aeropuertosantiago.cl/english/mediosdetransportedesdeyhaciascl/minibuses.html The bus (http://www.aeropuertosantiago.cl/english/mediosdetransportedesdeyhaciascl/busesyminibuses.html) can be found exiting the airport to the right and you should take it up to Metro Pajaritos. Take the metro in direction Los Dominicos and exit at metro Los Leones . Then use the following map For los españoles : For Apart Hotel Lyon: The bus costs 1500 or 1600 pesos (depending on the company) and the metro between 610720 pesos (depending on the time of the day). Practical informations on the conference:MAPThe conference will be held at Auditorio Ninoslav Bralic at Campus San Joaquin of PUC days 1,2, 6 and 7. Day 3 will be shared between PUC and CMM (we will provide transportation between the venues that day). The remaining days will be at CMM. To arrive to Auditorio Ninoslav Bralic: (THESE ARE THE DIRECTIONS FOR THE FIRST DAY!) From metro Los Leones take the metro in direction San Pablo, transfer at metro Baquedano to the green line (line 5) in direction Vicente Valdez. Get off at metro San Joaquín and enter the campus (that will be on your right when exiting the metro). From there walk straight to the chapel and turn right. Auditorio Ninoslav Bralic will be next to the astronomy department. If you want to come by taxi, you should indicate the driver to go to campus San Joaquin de Universidad Catolica, next to metro San Joaquin (al lado del metro San Joaquin). The ride should cost between 5000 and 10000 pesos, depending on the traffic and should take around half an hour.<\p> To arrive to CMM:<\p> From metro Los Leones take the metro in direction San Pablo, transfer at metro Los Heroes to the yellow line (line 2) in direction La Cistena. Get off at metro Toesca and follow this map:<\p> Alternatively, you can stay in the red line and get off at metro Republica and follow this map If you want to come by taxi, you should indicate the driver to go to Beaucheff 850 (Beaucheff con Blanco Encalada). The ride should cost between 5000 and 10000 pesos, depending on the traffic and should take around half an hour. Registration:PROLONGATION REGISTRATION AVAILABLE BY EMAIL WITH LAURENT HAUSWIRTH: "hauswirth(nospam)univmlv.fr"
