Flagfolds: multi-dimensional varifolds to handle discrete surfaces

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. Though flexible, varifolds require the knowledge of the dimension of the shape to be considered. We then embed all d-dimensional Grassmannians into symmetric positive semi definite matrices with trace 1, that we endow with a distance coinciding with the Riemannian one in each Grassmannians.

Building upon the aforementioned embedding of Grassmannians, we propose a generalization of varifolds, that we call flagfolds, in order to model multi-dimensional shapes. The notion of first variation extends to such flagfolds and we are investigating whether some form of Allard's rectifiability theorem extends as well.

Collaborators: C. Labourie, G.P. Leonardi, S. Masnou and X. Pennec.