BANFF Women in Geometry: Metric Geometry Group

BIRS Women in Geometry Workshop research group:

The Intrinsic Flat Convergence of Alexandrov Spaces

Our subgroup is considering the Intrinsic Flat Convergence of Alexandrov spaces. There is one immediate project in this area which we believe can be proven during the week of the workshop and then a number of possible further directions we may wish to explore. We will be creating a problem list for possible future collaborations. We invite you to suggest open problems and questions before the workshop begins, and these will be posted on this website.

The participants in our subgroup are:

Senior:

Catherine Searle (Alexandrov Geometry)

Christina Sormani (Convergence)

Junior:

Maree Jaramillo (Convergence and Smooth Metric Measure Spaces)

Raquel Perales (Convergence and Alexandrov Geometry)

Priyanka Rajan (Hopf Fibrations)

Anna Siffert (Harmonic maps, Cohomogeneity one Manifolds)

Resources:

Before the workshop we are all centering ourselves on the same basic background

so that we are using the same notation and vocabulary:

The key resources for metric geometry:

Everyone needs to know basic metric space and GH convergence theory:

[BBI] Burago-Burago-Ivanov's textbook on Metric Geometry is available online free:

http://www.math.psu.edu/petrunin/papers/alexandrov/bbi.pdf

The useful sections for our participants would be:

Chapters 1-4, Chapter 7 and Chapter 10

In fact, the participants will know most of Chapter 1-4, so really it is Chapter 7 and Chapter 10.

The key resources for Alexandrov Geometry:

A good survey of Alexandrov spaces can be found at

Shiohama's An Introduction to Alexandrov Spaces

as well as

Plaut's Metric Spaces of Curvature ge K

Key background in this area can be found in:

[AKP] S. Alexander, V.Kapovitch, A. Petrunin, Alexandrov geometry, available online free:

http://www.math.psu.edu/petrunin/papers/alexandrov-geometry/

The useful sections for our participants would be:

Sections 5, 6, and 8 for background. Also 13, 15, 17, 19, and 20. Useful tools in 23, 24, and 25.

Also we will need to study oriented Alexandrov spaces:

[HS] Harvey and Searle, C. "Orientation and Symmetries of Alexandrov Spaces" (arxiv post)

The key resources for Intrinsic Flat Convergence and Integral Current spaces:

The Geometry Festival presentation is a nice introduction:

"Intrinsic Flat Convergence" Geometry Festival 2008 (pdf file of this presentation) (video)

Then the original Sormani-Wenger JDG paper introducing

intrinsic flat convergence and integral current spaces is crucial:

• [SW-JDG] C. Sormani and S. Wenger, "The Intrinsic Flat Distance between Riemannian Manifolds and Integral Current Spaces" Journal of Differential Geometry, Vol 87, (2011) (arxiv preprint)

It builds upon work of Ambrosio-Kirchheim but provides the

background material from their work as needed. Nevertheless here is the Ambrosio-Kirchheim paper:

• [AK-Acta] L. Ambroio and B. Kirchheim, "Currents in Metric Spaces" Acta Math. 185 (2000), no. 1, 1--80. (preprint, reprint, mathscinet review)

The only paper concerning intrinsic flat convergence and Alexandrov curvature is a new arxiv preprint

of Li-Perales:

[Li-Perales-arxiv] N. Li and R. Perales, "On the Sormani-Wenger Intrinsic Flat Convergence of Alexandrov Spaces" arxiv preprint

All papers concerning intrinsic flat convergence are linked to

here (but are mostly unrelated to this workshop):

https://sites.google.com/site/intrinsicflatconvergence/

Resulting Preprint (after meeting on a few more occasions):

• "Alexandrov Spaces with Integral Current Structure"

  • by Maree Jaramillo, Raquel Perales, Priyanka Rajan, Catherine Searle, Anna Siffert arxiv preprint

  • Abstract: We endow each closed, orientable Alexandrov space (X,d) with an integral current T of weight equal to 1, $\partial T = 0 and \set(T) = X$, in other words, we prove that (X,d,T) is an integral current space with no boundary. Combining this result with a result of Li and Perales (arxiv preprint), we show that non-collapsing sequences of these spaces with uniform lower curvature and diameter bounds admit subsequences whose Gromov-Hausdorff and intrinsic flat limits agree. This builds upon a paper of Mitsuishi (arxiv preprint) which does not involve integral current spaces but does apply Ambrosio-Kirchheim theory to Alexandrov spaces.