MSRI-F13

In the first half of the seminar we will cover the key papers related to the weak convergence of manifolds:

• Gromov-Hausdorff Convergence following Burago-Burago-Ivanov's Textbook

• Sturm's Wasserstein Distance between Metric Measure spaces

• Currents on Metric Spaces as defined by Ambrosio-Kirchheim and their Compactness Theorem

• Intrinsic Flat Convergence as defined by Sormani-Wenger including Applications to General Relativity completed by Lee-Sormani

• Varifold convergence and the work of Almgren, Allard, and their compactness theorems.

In the second half of the semester, each meeting will cover a single paper selected by Professor Sormani that is relevant to the research of particular participants in the seminar. Please contact Professor Sormani (sormanic at gmail etc) if you would like to participate in this seminar.

Meetings:

• August 28: Gromov-Hausdorff and measured GH convergence. (Sormani) Basic definitions and the statement of Gromov's Compactness and Embedding Theorems. We will also discuss questions arising related to Gigli's introductory course from the Optimal Transport Workshop. See Burago-Burago-Ivanov's textbook 7.1-7.4.

• Sept 4: short organizational meeting 4-5pm to avoid conflict with MSRI Women in General Relativity Workshop (note that on Sept 5: Sturm presents the L^2 distortion distance at the OT Seminar and we recommend our participants to attend that seminar)

• Sept 11: Smooth Convergence away from Singularities and Gromov-Hausdorff Convergence (Sormani) We will discuss well known examples where a sequence of Riemannian manifolds converges smoothly away from a singularity and study whether the metric completions of the smooth limits agree with the Gromov-Hausdorff limits. We welcome new participants from the MSRI Intro to General Relativity Workshop and will review the definitions of Gromov-Hausdorff convergence and smooth convergence away from singularities.

• Sept 18: Federer-Fleming's Flat distance for Currents on Euclidean Space (Xin Zhou) [Morgan-Geometric Measure Theory, Lin&Yang-Geometric Measure Theory] This is an essential tool used to handle the development of singularites in hypersurfaces in Euclidean space and is applied in free boundary PDEs and also more recently related to the convergence of Riemannian manifolds. The singular hypersurfaces are called integral currents and are rectifiable. Federer-Fleming have a compactness theorem stating that sequences of surfaces with uniform upper bound on their volume and the volume of their boundaries that are located in a common compact region in Euclidean space have subsequences which converge in the flat or weak sense as currents to an integral current.

• Sept 25*: Ambrosio-Kirchheim Currents on Metric Spaces (Sajjad Lakzian) [AK-Acta] Here Ambrosio and Kirchheim have extended Federer-Fleming's work to define and study integral currents lying in a complete metric space. They extend the compactness theorem to this setting.

• Oct 2*: Sturm's Wasserstein Distance (Matthais Erbar) [Sturm-Acta] This is a distance between pairs of Riemannian manifolds (or more generally metric measure spaces) defined by taking an infimum over all isometric embeddings of the metric spaces into a common metric space Z and then taking the Wasserstein distance in Z between the push forwards of their metrics. Equivalent definitions will also be discussed. Sturm has a compactness theorem in this setting for metric measure spaces with doubling measures.

• Oct 9: Intrinsic Flat Convergence of Riemannian Manifolds (Sormani) [SW-JDG] This is a distance between Riemannian manifolds (or more generally integral current spaces which are metric spaces with an integral current structure defined as in Ambrosio Kirchheim) which is defined by taking an infimum over all isometric embeddings of the metric spaces into a common metric space and taking the flat distance between the push forwards of their integral current structures. Simple means of estimating this distance without needed the full theory of Ambrosio-Kirchheim currents will be discussed as well. Wenger's Compactness Theorem applies to sequences of Riemannian manifolds with a uniform upper bound on the volume, volume of the boundary and the diameter of the manifold.Someone may volunteer to read and present Wenger's Compactness Theorem later in the semester.

• Oct 16: Gromov's Compactness Theorem (Shuanjian Zhang) Also this week MSRI Numerical Optimal Transport Workshop.

• Monday Oct 21: Sormani speaks at Stanford

• Oct 23: Schoen-Yau Proof of the Positive Mass Theorem (Alessandro Carlotto) Here we will cover basic theorems about Gauss curvature of a stable minimal surface in M^3 with nonnegative scalar curvature and consequence of Gauss-Bonnet followed by the Schoen-Yau proof of the positive mass theorem with emphasis on the rigid situation when the ADM mass is 0.

• Oct 30*: Almgren's Theory of Varifolds and Allard's Regularity (Ling Xiao)

  • Nov 6: Simon's Varifold Regularity as applied by Eichmair to Jang's Equation (Anna Sakovich) We recall the origins of Jang's Equation and sketch how it was applied by Schoen and Yau to prove the spacetime positive mass theorem in dimension 3. We then discuss how methods of Geometric Measure Theory apply to prove existence and regularity of the geometric solutions in higher dimensions, based on the work of Eichmair.

• Nov 13*: Wenger's Compactness Theorem (Sajjad Lakzian) Wenger's Compactness Theorem states that a sequence of Riemannian manifolds with a uniform upper bound on volume and on the volume of the boundary and on diameter has a subsequence which converges in the Intrinsic Flat sense to an integral current space (possibly the 0 space). The proof uses techniques which may be useful towards proving similar compacntess theorems for Sturm's Wasserstein distance, D.

• Nov 20: Schoen-Yau Torus Rigidity and Bray-Brendle-Neves Splitting Theorem (Davi Maximo) These theorems concern three dimensional manifolds with nonnegative scalar curvature. Both apply the Schoen-Yau Theorem concerning stable minimal surfaces in such spaces. The proofs of these theorems as well as the Schoen-Yau Theorem will be presented.

• Nov 27*: Thanksgiving Eve, no meeting

• Dec 4: Applications of Intrinsic Flat Convergence to questions arising in General Relativity (Sormani) In this last minute substitution talk, the work of Dan Lee and Christina Sormani concerning sequences of asymptotically flat manifolds with nonnegative scalar curvature whose ADM mass converges to 0 will be presented. In the rotationally symmetric case an explicit construction has been made to demonstrate that in the rotationally symmetric case the sequence converges in the pointed intrinsic flat sense to Euclidean space. In general it is conjectured that the limit is Euclidean space. Additional special cases will be discussed.

• Dec 9: [Extra meeting Monday at 10:30 am] Cheeger-Colding Theory (Zahra Sinaei)

• Dec 11: Brakke Flow (Ling Xiao)

Active Participants: