Speaker

Yingying Zhang 张蓥莹

Time

Fridays, 9:50 am-12:15

Feb. 21 - Jun. 6, 2025

Venue

B626

Shuangqing Complex Building A

Course description

Elliptic and parabolic methods play an important role in the study of modern geometry and topology, typically, problems from geometry can be interpreted as non-linear partial differential equations due to the non-linear feature of the geometry, and solving these geometric equations need both analysis technique and good understanding of the geometric and topological structures of the manifold. Many cornerstone results in geometry are established by using elliptic and parabolic methods, for example, Hodge theorem, Plateau problem, Calabi conjecture, Poincare conjecture, (Hermitian) Yang-Mills, etc. In spring 2025, we plan to select the following topics to illustrate the analysis methods including the maximum principle, calculus of variation and parabolic flows.

1. Yau’s original proof of the Schwarz Lemma, gradient estimate of harmonic functions on complete manifolds, and the Calabi conjecture. These results are very nice application of the maximum principle methods, and techniques developed in these papers now become standard tools in geometric analysis.

2. Sack-Uhlenbeck’s proof on the existence of minimal 2-spheres in compact Riemannian manifolds. This paper adopts the method in the calculus of variation, the most important part in this paper is introducing the “blowing up analysis” to handle the “energy concentration” problem, which yields to the famous “bubbling phenomena” for harmonic maps. From then, “blowing up” analysis becomes a powerful tool in the study of formation of singularity in geometry.

3. Hamilton’s first paper on Ricci flow. The main result is to show a compact Riemannian 3-manifold with positive Ricci curvature must be diffeomorphic to a spherical space form. The “Ricci flow” was introduced in this paper to prove the result and later on becomes an essential tool to solve the Poincare conjecture.

We shall try to minimize the background in geometry and analysis as much as possible, but basic knowledge on Riemannian geometry and 2nd order linear elliptic PDE theory are encouraged to learn before the course, but we shall summarize key ingredients in the first few lectures. Students are expected to work on large amount of calculation to get a better feeling on how analysis goes on solving geometric problems. If time permits, we shall also include later development based on above three papers, either on application of the results or techniques.

Reference:

1. Yau, Shing-Tung, Harmonic functions on complete Riemannian manifolds. CPAM28, 1975, 201-228.

2. Yau, Shing-Tung, A general Schwarz lemma for Kahler manifolds. AJM, 100, 1978, no. 1, 197-203.

3. Yau, Shing-Tung, On the Ricci curvature of a compact Kahler manifold and the complex Monge-Ampere equation.I.CPAM 31, 1978, no.3. 339-411.

4. Sacks, J; Uhlenbeck, K. The existence of minimal immersions of 2-spheres. Annals(2), 113, 1981, no.1., 1-24.

5. Hamilton, Richard S, Three-manifolds with positive Ricci curvature. JDG 17, 1982, no.2. 255-306

Course requirement:

putting hands on doing calculation, raising questions frequently, discussion to learn are strongly encouraged.