Clay Research Institute on The Global Theory of Minimal Surfaces
Program for weeks one and two

The first two weeks of the 2001 Clay Mathematics Institute will include a graduate level introduction to the theory of minimal surfaces. There will be two or three lecture series and additional events and activities, including homework sessions, open problem discussions, demonstrations and instruction on computer graphics techniques. Attending will be graduate students from the MSRI sponsoring institutions and additional graduate students and researchers sponsored by the Clay Institute.

Attending students are nominated by an MSRI sponsor or nominated as a Clay Mathematics Institute participant via the methods indicated in the CMI Workshop Page at the Clay Research Institute on The Global Theory of Minimal Surfaces. For the main program see the MSRI Workshop Page for the Clay Research Institute on The Global Theory of Minimal Surfaces

Program for weeks one and two

Main Lecture Series - Topics

Frank Morgan will give nine lectures on the subject of Geometric Measure Theory and the Proof of the Double Bubble Conjecture:
Last year Hutchings, Morgan, Ritore and Ros announced a proof of the Double Bubble Conjecture, which says that the familiar standard double soap bubble provides the least-area way to enclose and separate two given volumes of air. It was only with the advent of geometric measure theory in the 1960s that mathematicians were ready to deal with such problems involving surfaces meeting along singularities in unpredictable ways. The lectures will discuss modern, measure-theoretic definitions of "surface," compactness of spaces of surfaces, and finally the proof of the double bubble conjecture. Homework will vary from basic exercises to open problems. The text Geometric Measure Theory: A Beginner's Guide (3rd edition) by Frank Morgan will be made available, as well as additional notes and materials. (Students nominated by MSRI sponsors will receive a copy of the book on arrival. Several copies will be available for use by other participants.) There will be sessions on exercises and on open problems.

Bill Meeks will give five lectures on geometric results in classical minimal surface theory.
Topics will include: Curvature(s) in surface theory; Gauss and mean curvature; First variation of area; Definition of a minimal surface (H=0); The theory of minimal surfaces; The Weierstrass Rep; Harmonic forms; Gauss map conformal; Surface data in terms of G and dh; Symmetries; Associate family; Conjugate surfaces; Symmetry lines; Schwarz reflection principle; A survey of unsolved problems in minimal surface theory and approaches for solving them.

John McCuan will run some sessions discussing background material used in the other lecture series.

Rick Schoen will give three lectures on applications of minimal surfaces to General Relativity and Riemannian Geometry

Konrad Polthier will give three lectures on Computational aspects and discrete minimal surfaces

Matthias Weber will give five lectures on computational aspects of minimal surface theory

Brian White will give three lectures, subject to be posted later.

Research lecture series

These talks will be an introduction to some topics of recent research, at an expository level.

Mohammad Ghomi
University of South Carolina
Topology of minimal surfaces with convex boundary
This talk will be a concise introduction to a conjecture of Meeks which states that a compact minimal surface bounded by a pair of convex curves in parallel planes has to be topologically an annulus. We will give a survey of partial results due to Schoen, Meeks and White, and Ross. Further we will discuss some approaches as to how to solve and how not to solve the conjecture, including the relation between the topology and umbilic points of such surfaces.

Nicolaos Kapouleas
Brown University
Singular perturbation constructions for minimal surfaces
The general idea of singular perturbation constructions will be briefly outlined, and then various constructions will be presented. Finally we will mention some open problems.

Harold Rosenberg
University of Paris 7
Periodic minimal surfaces
Examples of properly embedded periodic minimal surfaces, and the geometry and topology of finite topology (in the quotient) periodic surfaces. For example, the topological obstructions to the existence of properly embedded doubly-periodic minimal surfaces.

Hermann Karcher
Universitat Bonn
TBA

Francisco Martin
Universidad de Granada
Complete nonorientable minimal surfaces in R^3
An introduction to complete nonorientable minimal surfaces will be given: known examples, methods of construction, geometrical and topological properties. Some of the more interesting unsolved questions in this area will be presented: existence of examples with the lowest total curvature, problems related with the generalized Gauss map and stability.

Joaquin Perez
Universidad de Granada
TBA

Notice to students and workshop participants
Those of you who are interested in computational aspects of the subject and have access to laptop computers are encouraged to bring these to the workshop. If you have access to Mathematica through a site license, that will also be helpful.

Some related sites:

Interesting examples and graphical representations of minimal surfaces, and some expository material can be found at: