北京雁栖湖应用数学研究院

2024年02月27日 至 05月28日

Basic knowledge on Riemannian geometry and PDE.

1. Preliminaries: curvatures on Riemannian manifolds
2. Existence and uniqueness of the Ricci flow.
3. Hamilton’s maximum principle
4. Hamilton’s classic results in dimensions 2, 3 and 4
5. Differentiable sphere theorem

[1] B. Andrews, C. Hopper, The Ricci flow in Riemannian geometry: a complete proof of the differentiable 1/4-pinching sphere theorem, Lecture Notes in Mathematics, Springer, Heidelberg, (2011).
[2] C. Böhm, B. Wilking, Manifolds with positive curvature operator are space forms, Ann. of Math. 167 (2008) 1079-1097.
[3] S. Brendle, R. M. Schoen, Manifolds with 1/4-pinched curvature are space forms, J. Amer. Math. Soc. 22(1) (2009) 287-307.
[4] S. Brendle, Ricci flow and the sphere theorem, Graduate Studies in Mathematics, AMS Press, (2010).
[5] D. DeTurck, Deforming metrics in the direction of their Ricci tensors, J. Differ. Geom. 18 (1983) 157-162.
[6] R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differ. Geom. 17 (1982) 255-306.
[7] R. S. Hamilton, Four-manifolds with positive curvature operator, J. Differ. Geom. 24 (1986) 153-179.
[8] R. S. Hamilton, The Ricci flow on surfaces, Contemp. Math. 71 (1988) 237-261.
[9] W. X. Shi, Deforming the metric on complete Riemannian manifolds, J. Differ. Geom. 30 (1989) 223-301.

Undergraduate , Advanced Undergraduate , Graduate

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