Ecole d'été 2004: Surfaces minimales et problemes variationnels (30 juin au 8 juillet 2004)
Programme
Alexander Bobenko
* Discrete minimal surfaces made from circles and related variational problems
* 3 lectures
* We define discrete minimal surfaces made from circles and show how fundamental features of the classical theory (Weierstrass representation, variational description, symmetries, isometry preserving the Gauss map) persist in the discrete setup. Some other related variational problems (Willmore) are also discretized which leads also to some interesting problems of discrete geometry and applications.
* References : A.I. Bobenko, T. Hoffmann, B.A. Springborn, Minimal surfaces from circle patterns: Geometry from combinatorics, math.DG/0305184 http://www.arxiv.org/abs/math.DG/0305184
Ken Brakke
* Surface Evolver
* 3 lectures
* I will be giving an introductory talk about my Surface Evolver software, which models liquid surfaces with various kinds of energies and constraints. There will also be tutorial sessions at various levels arranged according to the interests of participants.
* [[http://www.susqu.edu/brakke/evolver|Evolver's home page]]
William Meeks
* The geometry and topology of properly embedded minimal surfaces in R3
* 4 lectures
* Title of talk 1: A survey of recent progress in the classical theory of minimal surfaces.
* Abstract: I will cover briefly many of the recent developments in the theory over the last decade. I will discuss the compactness and regularity results of Colding-Minicozzi and their applications to understanding the asymptotic behavior of properly embedded minimal surfaces of finite genus. I will discuss some of the important progress made on the conformal structure, topology and classification of properly embedded minimal surfaces, especially in the case when these surfaces have finite genus.
* Title of talk 2: The uniqueness of the helicoid, the asymptotic behavior of minimal surfaces of finite genus and topological obstructions.
* Abstract: I will give a sketch of the Meeks-Rosenberg proof of the uniqueness of the helicoid. Part of this proof involves a general theoretical result on minimal laminations of R^3, which I will discuss. I will try to indicate how the theory of Colding-Mionicozzi can be used to study properly embedded minimal surfaces of finite genus and to obtain related topological obstructions. In particular I will explain the usefulness of "blowing-up" a sequence of minimal surfaces on the scale of topology, in order to obtain deeper theoretical results.
* Title of talk 3: Applications of universal superharmonic functions to conformal and topological questions in minimal surface theory.
* Abstract: I will go over the theory of superharmonic functions as developed by Collin, Kusner, Meeks and Rosenberg to understand the conformal properties of proper minimal surfaces in R^3. Also I will give their proof that the middle ends of a properly embedded minimal surface are never simple ends. Finally I will explain how this last result is applied by Frohman and Meeks in their proof of the Topological Classification Theorem for minimal surfaces in R^3.
Franck Morgan
* Geometric Measure Theory and Isoperimetric Problems
* 4 lectures
* In 1884, Schwarz proved the classical isoperimetric theorem that a round ball provides the least-perimeter way to enclose given volume in R3. In 2002, Hutchings, Morgan, Ritore', and Ros proved that the familiar double soap bubble provides the least-perimeter way to enclose and separate two given volumes. Since double bubbles have singular curves, the proof requires geometric measure theory, which uses measure theory to generalize differential geometry to singular surfaces. The course will include an introduction to geometric measure theory, a discussion of isoperimetric problems (including double bubble problems and spaces with densities), and open questions. There are no special prerequisites.
* References :
* Geometric Measure Theory (2000 edition) by Frank Morgan.
* [[http://www.williams.edu/Mathematics/fmorgan|Morgan's home page]]
Frank Pacard
* Analytical aspects of minimal and constant mean curvature surfaces
* 4 lectures
* Abstract : We will explain the analytical framework needed to understand the moduli space theory for minimal surfaces and constant mean curvature surfaces. Based on this, we will describe the end-to-end connected sum construction for both compact and noncompact constant mean curvature surfaces. Finally, we will give a general framework for the connected sum of minimal or constant mean curvature surfaces. The end-addition procedure will also be described.
Antonio Ros
* The Periodic Isoperimetric Problem
* 4 lectures
* The Isoperimetric problem consists of the study of least surfaces under a volume constraint in different spaces (Euclidean, spherical,...) and different settings (prescribed symmetry, prescribed topology,...) and it is intimately related to surfaces with constant mean curvature. In your lectures we will put the emphasis in the case of periodic surfaces in Euclidean three-space. These surfaces have been proposed as the simplest mathematical model to explain certain shapes appearing in a number of interface phenomena with crystallographic symmetry.
Martin Traizet
* Construction of minimal surfaces
* 3 lectures
* When one tries to construct minimal surfaces using Weierstrass Representation, the difficult part is solving the so-called Period Problem. I will present a strategy to solve the Period Problem which reduces it to a set of algebraic (polynomial) equations. We will first see examples of situations where the method has been applied succesfully and has given interesting new examples of minimal surfaces (desingularisation of planes ; embedded minimal surfaces with no symmetries ; higher genus periodic helicoids). I will recall the needed tools from complex analysis (meromorphic forms on compact Riemann surfaces ; Riemann surfaces with nodes). Then we will see in more details how the method works.
Mike Wolf
* Teichmuller Theory and Minimal Surfaces
* 3 lectures
* We give an introduction to Teichmuller-theoretic techniques in minimal surface theory. We begin with a primer on Teichmuller theory (the analytic study of moduli spaces of Riemann surfaces), particularily its definition, how one does calculus in a moduli space of conformal structures, natural coordinates on Teichmuller space, flows on moduli spaces and methods of degeneration and regeneration. For applications to minimal surface theory, one needs to understand variations of conformal structures that preserve other geometric structures, so we give a quick introduction to singular flat surfaces; we then apply these techniques to prove existence, uniqueness and embeddedness theorems for minimal surfaces in a few contexts. If time permits, we discuss the related subject of harmonic maps of surfaces to complete manifolds.