In this course we take up two representative topics in geometric analysis, one is harmonic map and the other is Ricci flow. Harmonic maps are the generalization of harmonic functions, geodesics and minimal surfaces, is first developed in geometric analysis. Ricci flow is a nonlinear partial differential equation for Riemannian metrics introduced by Richard Hamilton in 1982 to find canonical metrics on manifolds. We explain basic results for both topics.
Through these two topics we understand how the method of analysis and the method of geometry are used in unified manner to solve problems in geometric analysis.
・Be familiar with the basic techniques in partial differential equations.
・Be familiar with many basic examples of harmonic maps and solutions of Ricci flow.
・Understand basic theory of harmonic maps and associated heat flow.
・Understand basic theory of Ricci flow.
Riemannian manifold, curvature tensor, geodesic, energy functional, harmonic map, heat equation, Einstein-Hilbert functional,
Ricci flow, Ricci soliton
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
This is a standard lecture course. There will be some assignments.
Course schedule | Required learning | |
---|---|---|
Class 1 | Basic concepts of Riemannian geometry | Details will be provided during each class session |
Class 2 | Length of curves and geodesics | Details will be provided during each class session |
Class 3 | Existence theorem of geodesics | Details will be provided during each class session |
Class 4 | The 1st variation formula of energy of curves | Details will be provided during each class session |
Class 5 | The 2nd variation formula of energy of curves | Details will be provided during each class session |
Class 6 | Applications to geometry of geodesics | Details will be provided during each class session |
Class 7 | Definitions and examples of harmonic maps | Details will be provided during each class session |
Class 8 | Existence theorem of harmonic maps | Details will be provided during each class session |
Class 9 | Applications to geometry of harmonic maps | Details will be provided during each class session |
Class 10 | Overview of Hamilton's Ricci flow ( definition, background and history ) | Details will be provided during each class session |
Class 11 | Various solutions of Ricci flow | Details will be provided during each class session |
Class 12 | Existence and uniqueness of short time solutions of Ricci flow | Details will be provided during each class session |
Class 13 | Evolution of curvature | Details will be provided during each class session |
Class 14 | Evolution of curvature | Details will be provided during each class session |
Class 15 | Applications to geometry of Ricci flow | Details will be provided during each class session |
none required
・B.Andrews-C.Hopper, The Ricci flow in Riemannian Geometry, Lect. Notes in Math. Vol. 2011, Springer, 2011
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manifold theory, calculus of several variables, functional analysis