Organizers: TszKiu 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 lowdimensional 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 DiracType 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 3Manifolds with Positive Scalar Curvature
[Li19] Li The Dihedral Rigidity Conjecture for nPrisms
[WXY] Wang, Xie, Yu On Gromov’s Dihedral Extremality and Rigidity Conjectures
Date 
Speaker 
Title and abstract 

Jan 25th 

Organizational Meeting 
Feb 1st 
Yipeng Wang 
The C^0 Limit Theorem for Scalar Curvature Show/hide AbstractsThe theory of taking limits of Riemannian manifolds is extensively studied given that certain curvature quantities are uniformly bounded from below. For instance, the theory of Alexandrov spaces is developed to study limit spaces with sectional curvature bounded from below, and similar situations apply to spaces with bounded Ricci curvature via CheegerColdingNaber Theory. However, the corresponding results on scalar curvature are still far from being understood. Recently, Gromov proposed to study spaces with positive scalar curvature in terms of polyhedra and use it to prove the compactness theorem. We will discuss a simpler proof proposed via Ricci flow introduced by Bamler. At the same time, we will motivate the corresponding rigidity conjecture and some related questions about scalar curvature.

Feb 8th 
Yipeng Wang 
More about the Dihedral Rigidity Problem Show/hide AbstractsWe will discuss more motivations for the dihedral rigidity problem and focus on some relations with general relativity. After that, I will sketch the proof of the conjecture for a large class of polyhedrons in dimension between 3 and 7 modulo the theory of capillary surfaces.

Feb 15th 
Jingbo Wan 
Elliptic Boundary Value Problem for DiracType Operators: Part 1 Show/hide AbstractsDirac Operator is a powerful tool to study positive scalar curvature. Concerning positive scalar curvature on a manifold with boundary, it’s natural to ask how to formulate a valid boundary value problem for Dirac operator (1st order elliptic). In fact, Dirichlet boundary condition, which is natural for Laplacian operator (2nd order elliptic), turns out to be too strong for 1st order elliptic operators. In this talk, we focus on Diractype operators, with principal symbols capturing the Clifford relation just like the usual Dirac operator, and discuss some basic materials to get ready for elliptic boundary value problems.
Reference: [BB11] [BB13]

Feb 22nd 
Tin Yau Tsang (UC Irvine) 
Foliation by Perturbation Show/hide AbstractsWe would study the metric perturbation technique by ChodoshEichmair. From this, one can form a foliation of prescribed mean curvature surface which provides rigidity under scalar curvature assumption.

Mar 1st 
Jingbo Wan 
Elliptic Boundary Value Problem for DiracType Operators: Part 2 Show/hide AbstractsLast time, we defined the differential operator on the boundary associated to the given Dirac type operator using the information of the symbols. Recall that such associated boundary operator is a selfadjoint elliptic first order operator defined on a closed manifold (the boundary), so we can make use of its spectrum (L^2 decomposition of sections defined on the boundary) to investigate what kind of boundary condition is natural for the Dirac type operator. Statements will be given and the ideas of the proof will be sketched, and some important examples will be discussed.
Reference: [BB11] [BB13]

Mar 8th 
No seminar (Department Colloquim) 

Mar 15th 
No seminar (Spring break) 

Mar 22nd 
Jingbo Wan 
Elliptic Boundary Value Problem for DiracType Operators: Part 3 Show/hide AbstractsLast time, we studied the hybrid Sobolev spaces using the spectrum of a boundary adapted operator. In particular, the hybrid Sobolev space $\check{H}(A)$ is the image of (extended) trace map on the Diracmaximal domain, and any boundary condition we will consider is a closed subspace of $\check{H}(A)$. Among all the boundary conditions, a special class called Delliptic boundary conditions will be the main subject we are discussing. To understand this Delliptic boundary condition, we will start with the famous APS condition, and try to explore based on that. Under these Delliptic boundary conditions, nice boundary regularity result is obtained. And we will also discuss many other examples which belong to the class of Delliptic boundary conditions.
Reference: [BB11] [BB13]

Mar 29th 
Yipeng Wang 
Eigenvalue of Schrödinger Equations Show/hide AbstractsI will discuss a theorem originally due to Professor Fefferman and Phong, which is about the estimate of the lowest eigenvalue of Schrödinger equations with weak regularity assumptions on the potentials. In particular, we will look at the situations where the potential is small in a Morrey norm and how this is applied to the dihedral rigidity problem in Professor Brendle's work.

Apr 5th 
No seminar (Department Colloquim) 

Apr 12th 
Jingbo Wan 
Isoperimetric surface technique & Volumne Comparison theorems Show/hide AbstractsIn this talk, we will discuss the isoperimetric surface technique that was developed by Professor Hubert Bray in his PhD thesis. In particular, we are going to use such technique to study volume comparison theorems for positively curved manifolds, which includes a new proof for the Bishop’ volume comparison theorem and also the “Bray’s football theorem” a volume comparison theorem for scalar curvature in the case that the metric is close to the standard S^3 metric. We remark that the rigid case of Bray’s football theorem can be found in a beautiful survey paper written by Professor S.Brendle. (But we might not have the time cover it)
Reference: H. Bray, PhD thesis S. Brendle, Rigidity phenomena involving scalar curvature

Apr 19th 
Jingbo Wan 
Isoperimetric surface technique & Penrose Inequality Show/hide AbstractsIn this talk, we will continue to discuss the isoperimetric surface technique that was developed by Professor Hubert Bray in his PhD thesis. This time, we are going to study the effect of nonnegativity of scalar curvature on a complete asymptotically flat threemanifold. In particular, we will use the isoperimetric surface technique to prove two conditional Penrose inequality for single/multiple horizons cases. In contrast to last time, we are dealing with a minimizing process in noncompact manifolds, so we need to address the existence of area minimizers with its volume constraint.
Reference: H. Bray, PhD thesis S. Brendle, Rigidity phenomena involving scalar curvature

Apr 26th 
Aaron Chow 
Rigidity of convex polytopes under the dominant energy condition Show/hide AbstractsAbstract: In this talk, we will discuss a rigidity theorem for initial data sets corresponding to convex polytopes, which adhere to the dominant energy condition. To demonstrate the theorem, we employ a technique that entails approximating the target polytope using smooth convex domains and subsequently tackling a boundary value problem involving Dirac operators on these specific domains. This is a followup work of Brendle’s recent result.
