The course covers some basic materials on geometric measure theory with a strong emphasis on the applications to geometric variational problems. In particular, the course prepare the students for studying the mean curvature flow in the framework of geometric measure theory called the Brakke flow in the subsequent course in the 4Q.
The classical Plateau problem asks the existence and regularity of area-minimizing surface spanning a given closed curve. In recent years, the mean curvature flow which is the dynamic counterpart, has been studied intensively from differential geometric point of view as well as more analytic variational point of view. Since the mean curvature flow gives rise to singularities, it is natural to formulate the generalized solutions in the setting of singular surfaces in the framework of geometric measure theory. The course starts with the basic measure theory and explains the standard notion such as countable rectifiable set and density of measure.
Students can: carry out proofs using basic covering argument in measure theory, understand proofs using basic properties of countably rectifiable set, understand the relationship between measures and the first variations geometrically and analytically, understand the first variation in the setting of varifold.
geometric measure theory, countably rectifiable set, first variations, area-formula, mean curvature
Intercultural skills | Communication skills | Specialist skills | Critical thinking skills | Practical and/or problem-solving skills |
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Standard lecture course
Course schedule | Required learning | |
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Class 1 | Introduction to measure theory | Details will be provided during each class session. |
Class 2 | Hausdorff measure | |
Class 3 | Lipschitz functions and Rademacher Theorem | |
Class 4 | Submanifolds | |
Class 5 | area-formula and first variations | |
Class 6 | Countably rectifiable set | |
Class 7 | the first variations of countably rectifiable set | |
Class 8 | varifold |
Introduction to geometric measure theory, Leon Simon
Measure theory and fine properties of functions, Lawrence C. Evans and Ronald F. Gariepy
Assignments (100%).
Lebesgue measure, surface theory
tonegawa[at]math.titech.ac.jp