2019 Advanced courses in Analysis C

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Tonegawa Yoshihiro 
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Course description and aims

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.

Student learning outcomes

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

Competencies that will be developed

Intercultural skills Communication skills Specialist skills Critical thinking skills Practical and/or problem-solving skills
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Class flow

Standard lecture course

Course schedule/Required learning

  Course schedule Required learning
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

Reference books, course materials, etc.

Measure theory and fine properties of functions, Lawrence C. Evans and Ronald F. Gariepy

Assessment criteria and methods

Assignments (100%).

Related courses

  • MTH.C408 : Advanced topics in Analysis D1

Prerequisites (i.e., required knowledge, skills, courses, etc.)

Lebesgue measure, surface theory

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