401-3830-69L Seminar on Minimal Surfaces
|Semester||Autumn Semester 2019|
|Language of instruction||English|
|Comment||The total number of students who may take this course for credit is limited to twenty; however further students are welcome to attend.|
|Abstract||This course is meant as an invitation to some key ideas and techniques in Geometric Analysis, with special emphasis on the theory of minimal surfaces. It is primarily conceived for advanced Bachelor or beginning Master students.|
|Objective||The goal of this course is to get a first introduction to minimal surfaces both in the Euclidean space and in Riemannian manifolds, and to see analytic tools in action to solve natural geometric problems.|
Students are guided through different types of references (standard monographs, surveys, research articles), encouraged to compare them and to critically prepare some expository work on a chosen topic.
This course takes the form of a working group, where interactions among students, and between students and instructor are especially encouraged.
|Content||The minimal surface equation, examples and basic questions. Parametrized surfaces, first variation of the area functional, different characterizations of minimality. The Gauss map, basic properties. The Douglas-Rado approach, basic existence results for the Plateau problem. Monotonicity formulae and applications, including the Farey-Milnor theorem on knotted curves. |
The second variation formula, stability and Morse index. The Bernstein problem and its solution in the two-dimensional case. Total curvature, curvature estimates and compactness theorems. Classification results for minimal surfaces of low Morse index.
|Literature||Three basic references that we will mostly refer to are the following ones:|
1) B. White, Lectures on minimal surface theory, Geometric analysis, 387–438, IAS/Park City Math. Ser., 22, Amer. Math. Soc., Providence, RI, 2016.
2) T. Colding, W. Minicozzi, A course in minimal surfaces. Graduate Studies in Mathematics, 121. American Mathematical Society, Providence, RI, 2011. xii+313 pp.
3) R. Osserman, A survey of minimal surfaces. Second edition. Dover Publications, Inc., New York, 1986. vi+207 pp.
Further, more specific references will be listed during the first two introductory lectures.
|Prerequisites / Notice||The content of the first two years of the Bachelor program in Mathematics, in particular all courses in Real and Complex Analysis, Measure Theory, Topology.|
Some familiarity with the language of Differential Geometry, although not a formal pre-requisite, might be highly helpful. Finally, a first course on elliptic equations (especially on basic topics like Schauder estimates and the maximum principle) might also be a plus.