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- Academic unit or major
- Mathematics
- Instructor(s)
- Tonegawa Yoshihiro
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Fri3-4(H137)
- Group
- -
- Course number
- ZUA.C334
- Credits
- 1
- Academic year
- 2019
- Offered quarter
- 4Q
- Syllabus updated
- 2019/3/18
- Lecture notes updated
- -
- Language used
- English
- Access Index
-
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.
Keywords
mean curvature flow, Brakke flow, tangent flow, compactness theorem, existence theory, regularity theory
Competencies that will be developed
| ✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
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 |
Textbook(s)
none
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
Contact information (e-mail and phone) Notice : Please replace from "[at]" to "@"(half-width character).
tonegawa[at]math.titech.ac.jp
