ActividadesActivities
MinicursosMini-courses
- YanYan Li (Rutgers University, EEUUUSA)
Título:Title: On a fully nonlinear version of the Yamabe problem
Duración:Duration: 3.75h.Resumen:Abstract: We will first recall the Yamabe problem and its solution. Then we will study a fully nonlinear version of the Yamabe problem. In particular we will present some closely related Liouville theorems, apriori estimates, and existence and compactness of solutions.
- William H. Meeks III (University of Massachusetts at Amherst, EEUUUSA)
Título:Title: Constant mean curvature surfaces in homogeneous 3-manifolds
Duración:Duration: 3.75h.Resumen:Abstract: I will present some of the basic results on the geometry of constant mean curvature H≥0 surfaces M in a complete homogeneous 3-manifold X; such an M will be called an H-surface. We will call a foliation F of X a CMC foliation if all of the leaves of F are H-surfaces with H possibly varying. The key results in the mini-course will include:
- General theory of H-surfaces M in Riemannian 3-manifolds with an emphasis on the case M is complete and embedded, including work described in papers of Colding-Minicozzi and Meeks-Perez-Ros and Meeks-Tinaglia.
- Curvature estimates for CMC foliations of X. Based on joint work with Perez and Ros.
- Curvature estimates for H-disks with H>0 in X. Based on joint work with Tinaglia.
- Uniqueness results for H-spheres in X, which generalize the classical result of Hopf that for each H>0, there is a unique H-sphere in R3. Based on joint work with Mira, Perez and Ros.
- Frank Pacard (Université Paris 12, FranciaFrance)
Título:Title: Geometric aspects of the semilinear elliptic PDE's
Duración:Duration: 3.75h.Resumen:Abstract: I will describe some constructions of solutions to some semilinear elliptic equations, all of which are based on the understanding of minimal and constant mean curvature surfaces in Euclidean space.
For example, I will explain the role of minimal surfaces in the construction of entire solutions of the Allen-Cahn equation in ℝn, I will also present the construction of solutions for some overdetermined elliptic problems which arise in the study of extremal domains for the first eigenvalue of the Laplacian.
Descargar notas en formato pdfDownload notes in pdf format
- Harold Rosenberg (Instituto Nacional de Matemática Pura e Aplicada, BrasilBrazil)
Título:Title: Introduction to the work of Colding-Minicozzi on minimal surfaces in R3
Duración:Duration: 3.75h.Resumen:Abstract: Colding and Minicozzi have studied the possible limits of simply connected minimal surfaces in R3. I will present the beginning of their work in detail, and go as far as time permits.
ConferenciasTalks
- Pablo Mira (Universidad Politécnica de Cartagena, EspañaSpain)
Título:Title: Geometric PDE's in the presence of isolated singularities
Resumen:Abstract: A classical problem in the regularity theory of geometric PDEs concerns the study of regular solutions on a punctured disc. For some cases such as the minimal surface equation or other much more general elliptic geometric equations in divergence form, isolated singularities are automatically removable. This is also true for suitable concepts of generalized solutions of many other elliptic PDEs.
Nonetheless, some geometric PDEs like the elliptic Monge-Ampère equation describing graphs of constant curvature K >0 in ℝ³, i.e.
uxx uyy - uxy2 = K (1 + ux2 + uy2)2, K > 0
admit non-removable conical singularities. That is, they admit solutions on a punctured disc that extend continuously but not C1-smoothly to the puncture. For them, the usual regularity theory of generalized solutions is not applicable.
In this talk we shall explain how to deal with geometric PDEs in presence of non-removable isolated singularities. Using methods from complex analysis, geometry and elliptic PDEs, we will provide a local classification result in terms of the limit normal cone at the singularity. In the global case, we shall expose different classification results for solutions of special geometric PDEs in the presence of an arbitrary number of conical singularities. Descargar notas en formato pdfDownload notes in pdf format
- Robert Neel (Lehigh University, EEUUUSA)
Título:Title: Brownian motion on minimal surfaces
Resumen:Abstract: We introduce Brownian motion on minimal surfaces, including its relationship to the conformal structure of the surface and the ambient geometry of ℝ³. We then discuss applications to weak halfspace theorems and to showing parabolicity and quadratic area growth for ends of minimal surfaces constrained to lie in various regions.
Descargar notas en formato pdfDownload notes in pdf format
- Giuseppe Tinaglia (King's College London, Reino UnidoUK)
Título:Title: Curvature estimates and applications
Resumen:Abstract: In this talk I will discuss the significance of curvature estimates for constant mean curvature surfaces, present some classical ones and their applications.
Descargar notas en formato pdfDownload notes in pdf format
HorariosSchedule
| Lun 28Mon 28 | Mar 29Tue 29 | Mié 30Wed 30 | Jue 1Thu 1 | Vie 2Fri 2 | |
| 9:15-9:30 | AperturaOpening | ||||
| 9:30-10:45 | Rosenberg | Rosenberg | Rosenberg | Meeks | Pacard |
| 11:00-12:15 | YanYan Li | YanYan Li | YanYan Li | Pacard | Meeks |
| COMIDALUNCH | |||||
| 16:00-17:15 | Meeks | Pacard | Tinaglia | Neel | |
| 17:15-18:30 | Mira |
