Dos días de Análisis Geométrico en IEMath-GR

Event Details

Some colleagues, former students and friends of Antonio Ros have organized a couple of days of informal talks on Geometric Analysis, as a little homage to him on the occassion of his 60th birthday (and the 70th of Bill Meeks, who is currently on a stay in the IEMath-GR).

Program:

Thursday, June 1st

9:30 - 10:30 Francisco Urbano (Granada University): The mathematical contribution of Antonio Ros

10:30 - 11:30 Martin Traizet (Université François Rabelais Tours): Opening nodes and constant mean curvature surfaces

11:30 - 12:00 Coffee break

12:00 - 13:00 Rafael López (Granada University): New and many examples of minimal surfaces

Lunch

16:00 - 17:00 Frank Morgan (Williams College): Isoperimetric Problems

17:00 - 18:00 Francisco J. López (Granada University): Approximation theory by minimal surfaces and applications

21:00 Homage Dinner

Friday, June 2nd

9:30 - 10:30 William H. Meeks (Univ. of Massachussetts at Amherst): Recent progress in the theory of CMC surfaces in 3-manifolds

10:30 - 11:30 Laurent Hauswirth (Université de Marne-la-Vallée): TBA

11:30 - 12:00 Coffee break

12:00 - 13:00 David Ruiz (Granada University): Overdetermined elliptic problems and a conjecture of Berestycki, Caffarelli and Nirenberg

Lunch

16:00 - 17:00 Vicente Miquel (Valencia University): Type I singularities in the curve shortening flow associated to a density

17:00 - 18:00 Harold Rosenberg (IMPA): : Continuity properties of the A(1) function on the space of complete hyperbolic 3-manifolds of finite volume

Abstracts

• BILL MEEKS, Recent progress in the theory of CMC surfaces in 3-manifolds.

In this talk I will report on some of the new results in the theory of constant mean curvature (CMC) surfaces M in Riemannian 3-manifolds N. I will mention just a few of the topics touched on in this talk.

I first begin with the recent classification of CMC spheres in a homogeneous 3-manifold N and a sketch of its proof. The main result states that any two spheres in N with the same absolute mean curvature differ by an ambient isometry of N. Furthermore, the range of values of the mean curvature spheres can be described in terms of the geometry of the universal cover X of N: in the case that X is diffeomorphic to R^3, then there exists a sphere of constant mean curvature H in N iff H is greater than half the Cheeger constant of X and otherwise there exists a sphere of constant mean curvature in N for every real number. These results generalize previous work of Hopf, of Abresch-Rosenberg and more recently, of Danieil-Mira and of Meeks in the case of the Sol geometry.

In jointly with Tinaglia, we obtain curvature for embedded disks of fixed constant mean curvature H >0 in any fixed homogeneous 3-manifold. In the R^3 setting, this result implies that any complete embedded finite topology surface in R^3 of constant mean curvature is proper; this generalizes the previous work of Colding -Minicozzi in the case of minimal surfaces. Previous classification results then imply that the only complete embedded simply connected constant mean curvature surfaces in R^3 are the plane, the helicoid and round spheres. Another application of this work by Meeks-Tinaglia is to prove that complete embedded CMC surfaces of finite topology in a complete hyperbolic 3-manifold are proper if the mean curvature H is at least 1. On the other hand, Coskunuzer-Meeks-Tinaglia recently constructed for any H in [0,1) a non-proper, complete, stable embedded plane in hyperbolic 3-space having constant mean curvature H.

In 1982, Choi and Wang proved that an embedded closed minimal surface F in the the round three-sphere S^3 has a bound on its area that only depends on its genus; actually their result generalizes from the ambient space S^3 to any closed 3-manifold M with positive Ricci curvature. This result was then used by Choi and Schoen to prove the compactness of the moduli space of such examples of fixed genus g in M. Tinaglia and I have been able to give the following related result in the case of connected closed surfaces M embedded in any Riemannian homology 3- sphere manifold N:

Theorem: For any H>0 and non-negative integer g, there exists a constant A(N,g, H) such that any closed surface embedded in M of genus g and constant mean curvature H has area at most A(N,g,H).

This area estimate lead to a natural compactification of the moduli space of all such embedded constant mean curvature H examples in N with genus at most g, when H lies in a fixed compact interval [a,b] of positive numbers, and under a compact deformation of the Riemannian metric on N.

The recent classification of properly embedded minimal surfaces of genus 0 in R^3 given by Meeks-Perez-Ros, Lopez-Ros, Colin and of Meeks-Rosenberg play a role in the above area estimates, as do the curvature estimates of Meeks-Tinaglia for certain complete embedded CMC surfaces in a Riemannian 3-manifold.

At the end of my talk I will present a brief survey of some recent results on the existence and classification of CMC foliations of closed and non-closed 3-manifolds.

• DAVID RUIZ, Overdetermined elliptic problems and a conjecture of Berestycki, Caffarelli and Nirenberg.

In this talk we consider an elliptic semilinear problem under overdetermined boundary conditions: the solution vanishes at the boundary and the normal derivative is constant. These problems appear in many contexts, as in the study of free boundaries and obstacle problems. Here the task is to understand for which domains (called extremal domains) we may have a solution. This question has shown a certain parallelism with the theory of CMC surfaces, and also with the well-known De Giorgi conjecture for the Allen-Cahn equation.

The case of bounded extremal domains was completely solved by J. Serrin in 1971, and the ball is the unique such domain. Instead, the case of unbounded domains is far from being completely understood. In this talk we give a rigidity result in dimension 2, and also a construction of a nontrivial extremal domain in the form of a exterior domain.

This is joint work with Antonio Ros (U. Granada) and Pieralberto Sicbaldi (U. Granada and U. Aix Marseille).

• FRANCISCO J. LÓPEZ, Approximation theory by minimal surfaces and applications.

In this lecture we survey the theory of approximation by minimal surfaces and review some of its applications.

• RAFAEL LÓPEZ, New and many examples of minimal surfaces.

The Björling problem consists of finding a minimal surface containing a given curve and a prescribed unit normal vector to the surface along this curve. Under holomorphic assumptions, H. A. Schwarz proved local existence of such a surface obtaining an expression of the parametrization of the minimal surface involving nothing but integrals and analytic continuation of the initial data. Although this parametrization is simple, only a few number of explicit parametrizations of minimal surfaces are known in the literature. In this talk we provide new examples of minimal surfaces by solving the Björling problem for a large class of curves.

• FRANK MORGAN, Isoperimetric Problems.

Mathematicians from Archimedes to Ros have studied myriad isoperimetric problems with multitudinous applications. My talk will include recent results and open questions in various spaces.

• HAROLD ROSENBERG , Continuity properties of the A(1) function on the space of complete hyperbolic 3-manifolds of finite volume.
(Joint work with Laurent Mazet)

A(1)(M) is the least area of closed embedded minimal surfaces in M (counted with multiplicity 2 if realized by a non-orientable surface).

We describe the behavior of this function when a sequence of complete hyperbolic 3- manifolds of finite volume converges in the geometric topology to another such hyperbolic 3-manifold.

• MARTIN TRAIZET, Opening nodes and constant mean curvature surfaces.

Constant mean curvature surfaces in euclidean space have a Weierstrass-type representation called the DPW method. Opening nodes is a model for Riemann surfaces with small necks. We will combine the two theories to carry out gluing constructions for constant mean curvature surfaces.

• VICENTE MIQUEL, Type I singularities in the curve shortening flow associated to a density.
We define Type I singularities for the mean curvature flow associated to a density $\psi$ (\DMCF) and describe the blow-up at any singular time of these singularities. Special attention is paid to the case where the singularity comes from the part of the $\psi$-curvature due to the density. We describe a family of curves whose evolution under \DMCF (in a Riemannian surface of non-negative curvature with a density that is singular at a geodesic of the surface) produces only Type I singularities and study the limits of their rescalings.