Flagfolds: multi-dimensional varifolds to handle discrete surfaces

by Blanche Buet (Paris-Saclay)

Monday May 19, 2025, 2:00 PM → 3:00 PM Europe/Paris

Amphi Schwartz

We propose a natural framework for the study of surfaces and their different discretizations based on varifolds. Varifolds have been introduced by Almgren to carry out the study of minimal surfaces. Though mainly used in the context of rectifiable sets, they turn out to be well suited to the study of discrete type objects as well. While the structure of varifold is flexible enough to adapt to both regular and discrete objects, it allows to define variational notions of mean curvature and second fundamental form based on the divergence theorem.

Thanks to a regularization of these weak formulations, we propose a notion of discrete curvature (actually a family of discrete curvatures associated with a regularization scale) relying only on the varifold structure. We performed numerical computations of mean curvature and Gaussian curvature on 3D point clouds to illustrate this approach. Though flexible, varifolds require the knowledge of the dimension of the shape to be considered. By interpreting the product of the Principal Component Analysis, that is the covariance matrix, as a sequence of nested subspaces naturally coming with weights according to the level of approximation they provide, we are able to embed all d-dimensional Grassmannians into a stratified space of covariance matrices.

Building upon the proposed embedding of Grassmannians into the space of covariance matrices, we generalize the concept of varifolds to what we call flagfolds in order to model multi-dimensional shapes.

The first part of the talk will be dedicated to introducing varifolds (providing definition and examples, we will not need any regularity theory) and explaining how they can be used to model discrete surfaces and approximate their curvature.

In the second part, we will focus on multi-dimensional varifolds (flagfolds) that rely on the embedding of Grassmannians of different dimensions into a common stratified space.

Joint works with G.P. Leonardi, S. Masnou and X. Pennec.