2019 Advanced courses in Analysis D

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Tonegawa Yoshihiro 
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Course description and aims

The course introduces the mean curvature flow in the framework of geometric measure theory called the Brakke flow. Starting from the definition, the up-to-date research results will be presented.

A time-parametrized family of surfaces is called the mean curvature flow if the velocity of surface is equal to the mean curvature at each time and point. The mean curvature flow is a gradient flow of the surface area and its static counterpart is the minimal surface. Since the mean curvature flow has singularities, it is natural to consider the solution in the class of surfaces admitting singularities. Such class of surfaces can be expressed using the notion of varifold in geometric measure theory. In this course, starting from the definition of the Brakke flow, up-to-date research results are presented.

Student learning outcomes

Students can: understand the definition of Brakke flow, understand the basic properties and can generalize the notion to more general flow, obtain some basic existence and regularity theories and can extend them to more general settings.


mean curvature flow, Brakke flow, tangent flow, compactness theorem, existence theory, regularity theory

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 smooth mean curvature flow Details will be provided during each class session.
Class 2 definition of Brakke flow
Class 3 Huisken's monotonicity formula
Class 4 compactness theorem and tangent flow
Class 5 existence theorem for Brakke flow, I
Class 6 existence theorem for Brakke flow, II
Class 7 regularity theorem for Brakke flow



Reference books, course materials, etc.

Brakke's mean curvature flow: an introduction, Yoshihiro Tonegawa

Assessment criteria and methods

Reports (100%).

Related courses

  • ZUA.C333 : Advanced courses in Analysis C

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

Advanced courses in Analysis C

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