Fall 2026
Yanyan Li
Subtitle:
Yamabe Problem and Fully Nonlinear Conformally Invariant Equations
Course Description:
This course will focus on the Yamabe problem and its fully nonlinear conformally invariant extensions, including the \sigma_k-Yamabe problem in conformal geometry. We will begin with the classical Yamabe problem, including an outline of its proof and the main analytical ideas involved in its resolution. Along the way, we will study important tools surrounding the Yamabe equation, such as Liouville-type theorems, blow-up analysis, subcritical approximations, critical exponent equations, uniqueness results, compactness theorems, and examples of noncompactness.
The second part of the course will be devoted to fully nonlinear conformally invariant equations of second order, especially those involving the \sigma_k-curvature. Topics will include the \sigma_k-Yamabe problem, the \sigma_k-Nirenberg problem, and the \sigma_k-Loewner-Nirenberg problem. We will discuss analytic techniques for these equations, including a priori estimates, Liouville-type theorems, removable singularity results, and related compactness and blow-up phenomena.
The goal of the course is to introduce students to significant problems in conformal geometry and nonlinear elliptic PDE, with an emphasis on the analytical methods developed in the study of the Yamabe problem and its fully nonlinear counterparts. Open problems will be raised and discussed.
Text:
No textbook
Prerequisites:
Math 517 or permission of the instructor.
Schedule of Sections:
16:640:509 Schedule of Classes
Previous Semesters:
- Fall 2026 - Li, Yanyan
- Fall 2024 - Tobasco, Ian
- Spring 2024 - Carlen, Eric
- Fall 2023 - Li, Yanyan
- Fall 2022 - Li, Yanyan
- Fall 2021 - Li, Yanyan
- Spring 2021 - Sesum, Natasa
- Fall 2019 Prof. Y. Li