The Intrinsic Flat Distance between Riemannian Manifolds and other Integral Current Spaces
By C. Sormani, S. Wenger. 2010
121 citations recorded by the source corpus.
Inspired by the Gromov-Hausdorff distance, we define the intrinsic flat distance between oriented $m$ dimensional Riemannian manifolds with boundary by isometrically embedding the manifolds into a common metric space, measuring the flat distance between them and taking an infimum over all isometric embeddings and all common metric spaces. This is made rigorous by applying Ambrosio-Kirchheim's extension of Federer-Fleming's notion of integral currents to arbitrary metric spaces. We prove the intrinsic flat distance between two compact oriented Riemannian manifolds is zero iff they have an orientation preserving isometry between them. Using the theory of Ambrosio-Kirchheim, we study converging sequences of manifolds and their limits, which are in a class of metric spaces that we call integral current spaces. We describe the properties of such spaces including the fact that they are countably $\mathcal{H}^m$ rectifiable spaces and present numerous examples.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/18fe7c8f81cc37b220aebaa5be43445cd16faca8
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In Applied Mathematics
- Andrea Mondino
- Nicola Gigli
- Giuseppe Savar'e
- Luigi Ambrosio
- Sajjad Lakzian
- Fabio Cavalletti
- Elia Brué
- C. Sormani
In Geometry and Topology
- Nicola Gigli
- Andrea Mondino
- Giuseppe Savar'e
- Sajjad Lakzian
- C. Sormani
- Luigi Ambrosio
- Filippo Santambrogio
- C. Ketterer
Citations in the corpus
- Smooth Convergence Away from Singular Sets (2012, methodology): “In order to control the limits of such manifolds using only conditions on volumes, we apply techniques developed by the second author with Stefan and Wenger in [18] and [19].”
- Convergence of pointed non‐compact metric measure spaces and stability of Ricci curvature bounds and heat flows (2013, background): “spaces (for a different approach in the case of counting H N -rectifiable spaces, see [41]); here we briefly recall the main ideas; for the details the reader is referred to Subsection 3.”