Student Geometric & Analysis Seminar (Spring 2023): Scalar Curvature and Dihedral Rigidity

Organizers: Tsz-Kiu Aaron Chow, Jingbo Wan, Yipeng Wang

Time: Wednesdays 3:10 pm—5:10 pm

Location:  Room 528 at Columbia Math Department

This seminar dedicates to providing a survey about some known results and open questions related to the structure of Riemannian manifolds with scalar curvature bounded from below. Such manifolds seem could be characterized using metric geometric objects such as polyhedron comparison arguments via dihedral angles between adjacent weakly mean convex faces.  And those phenomena motivate the program proposed by Gromov, to study the delicate effect of scalar curvature on Riemannian manifolds through the shape of more singular metric spaces, namely the Dihedral Rigidity Conjecture. At the same time, we will introduce various analytic techniques such as constructing free boundary minimal hypersurfaces, establishing index theory for manifolds with corners, and solving certain elliptic boundary value problems. Applications in low-dimensional topology and mathematical physics will also be discussed if time permits. 

References: 

[B] Brendle Scalar Curvature Rigidity for Convex Polytopes

[BB11] B ̈ar, Ballmann Boundary Value Problems for Elliptic Differential Operators of First Order

[BB13] B ̈ar, Ballmann Guide to Boundary Value Problems for Dirac-Type Operators 

[G14] Gromov Dirac and Plateau Billiards in Domain with Corners

[G19] Gromov Four Lectures on Scalar Curvature

[G22] Gromov Convex Polytopes, Dihedral Angles, Mean Curvature, and Scalar Curvature 

[GL] Gromov, Lawson Spin and Scalar Curvature

[Li17] Li A Polyhedron Comparison Theorem for 3-Manifolds with Positive Scalar Curvature 

[Li19] Li The Dihedral Rigidity Conjecture for n-Prisms 

[WXY] Wang, Xie, Yu On Gromov’s Dihedral Extremality and Rigidity Conjectures

Title and Abstract (Spring 2023)