Online Mini-course: Convergence of Riemannian Manifolds with Lower Scalar Curvature Bounds

• This is a two weeks long course about the convergence of Riemannian manifolds with lowe scalar curvature bounds.

• Talks are conducted online with the zoom software (meeting ID: 387 384 3920, password: d8g6tm ).
• The time for each class is 9:30AM (Brussels time) or 4:30PM (Beijing time), from 3 November to 7 November (the first week), 12 November to 14 November (the second week), 17 November to 19 November (the third week).

• Edward Bryden (main lecturer, etbryden@gmail.com): Introduction
• Lizhi Chen (chenlzh@lzu.edu.cn)

Update on Seminar Schedule

Wednesday's (11/19) class is   cancelled   due to a time conflict.

The topic of ``Positive Mass Theorem'' will be included in the course notes. Monday's class (11/17) will be the final session of our mini-course.

There is no class on Tuesday (11/18).

Class on Wednesday (11/19) starts as usual at 16:30.

Class on Monday (11/17) starts as usual at 16:30.

There are no classes on Monday (November 10, 2025) and Tuesday (November 11, 2025).

Class on Wednesday (11/12), Thursday (11/13) , Friday (11/14) start as usual at 16:30.

There is no class on Thursday, November 6, 2025.

Class on Friday (11/7) starts as usual at 16:30.

Course notes will be uploaded later.

Syllabus

Week 1

1. Preliminary: Sobolev-Neumann constants, Gromov-Hausdorff convergence, Convergence in the Dong-Song sense, Stable systoles, Harmonic representation of integral forms
2. Scalar Curvature and Related Theorems

Week 2

1. Torus rigidity theorem
2. Stern's inequality

Week 3

1. Stability of Three-tori with small negative scalar curvature
2. Positive mass theorem rigidity using BKKS mass inequality
3. Dong-Song's Stability of Positive Mass Theorem

Course Notes and Recordings

Week 1

1. November 03, Riemannian Manifolds and the Sobolev-Neumann Constants : Slides , Recording , Notes
2. November 04, Gromov-Hausdorff convergence : Slides , Recording ,
3. November 05, de Rham Cohomology and Harmonic Maps : Slides , Recording ,
4. November 06, No Class
5. November 07, Poincare duality and the Hodge star map : Slides , Recording ,

Week 2

1. November 10, No Class
2. November 11, No Class
3. November 12, Curvature, Coarea, and Stern's inequality : Slides , Recording
4. November 13, Rigidity and Stability of Three-tori : Slides , Recording
5. November 14, Obtaining $L^3$ Estimates from Stern's Inequality and Integration by Parts : Slides , Recording ,

Week 3

1. November 17, Small Negative Curvature Implies Almost Constant Innerproducts : Slides , Recording ,
2. November 18, No Class
3. November 19, No Class

Questions and Discussions

We hope there are more discussions. You are very welcome to send your questions to any of the course organizers by Email: Lizhi (chenlzh@lzu.edu.cn), Edward (etbryden@gmail.com).
You can also choose to ask questions online during everyday's lecture time. We may post some of the discussions here.

Reading for the Prerequisite Knowledge

Books on Differentiable Manifolds and Riemannian Geometry

1. Loring W. Tu, 'An Introduction to Manifolds', Springer, 2011.
2. John M. Lee, ‘Introduction to Smooth Manifolds’, second edition, Graduate Texts in Mathematics 218, Springer, 2013.
3. Manfredo P. do Carmo, ‘Riemannian Geometry', Birkhäuser Boston, MA, 1992.
4. Peter Petersen, 'Riemannian Geometry', Springer Cham, third edition, 2016.
5. Sylvestre Gallot, Dominique Hulin and Jacques Lafontaine, 'Riemannian Geometry', Springer Berlin, Heidelberg, 2004.

References for the First Week's Classes

1. Peter Li, Geometric Analysis. Cambridge Stud. Adv. Math., 134. Cambridge University Press, Cambridge, 2012.
2. Dmitri Burago, Yuri Burago and Sergei Ivanov, A course in Metric Geometry. Graduate Studies in Mathematics, Volume 33, American Mathematical Society, 2001.
3. Raoul Bott, Loring W. Tu, Differential Forms in Algebraic Topology. Graduate Texts in Mathematics, Volume 82. Springer New York, NY, 1982.
4. Georges de Rham, Differentiable Manifolds, Forms, Currents, Harmonic Forms. Grundlehren der mathematischen Wissenschaften. Springer Berlin, Heidelberg, 1982.

References for the Second Week's Classes

1. Daniel L. Stern, Scalar curvature and harmonic maps to $S^1$, J. Differential Geom. 122 (2022), no. 2, 259--269.
2. Richard Schoen and Shing-Tung Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2) 110 (1979), no. 1, 127–142.
3. Mikhael Gromov and H. Blaine Lawson, Jr., Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Inst. Hautes Etudes Sci. Publ. Math. (1983), no. 58, 83–196.
4. H. Blaine Lawson Jr., Marie-Louise Michelsohn, Spin Geometry, in: Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989.
5. Jean-Pierre Bourguignon, Oussama Hijazi, Jean-Louis Milhorat, Andrei Moroianu and Sergiu Moroianu, A Spinorial Approach to Riemannian and Conformal Geometry. European Mathematical Society, 2015.
6. John William Scott Cassels, An introduction to the geometry of numbers, Springer Science & Business Media, 2012. (For a reference of brief introduction of flat tori and lattices, also see: Sylvestre Gallot, Dominique Hulin and Jacques Lafontaine, 'Riemannian Geometry', Springer Berlin, Heidelberg, 2004. )
7. Mikhail Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces. Birkhauser Boston, MA, 2007.
8. Isaac Chavel, Riemannian Geometry, A Modern Introduction. Second edition. Cambridge University Press, 2010.
9. Allen Hatcher, Algebraic Topology. Cambridge University Press, 2002. Download this book from author's webpage: PDF