BV functions and sets of finite perimeter in SR manifolds
with Luigi Ambrosio and Valentino Magnani
1. BV functions on manifolds
Definition 1 Let be an open set in . An function on is BV if
To make sense of this definition on a manifold, one merely needs to define divergence, i.e. a volume element suffices, and a class of admissible vector fields.
Definition 2 Let be a smooth oriented manifold, a smooth nowhere vanishing top dimensional form on , a compactly supported vector field on . The divergence of is the function such that
Definition 3 Let be a smooth function which is quadratic on fibers. Let denote the class of vector fields on such that everywhere.
Definition 4 Let be a smooth oriented manifold, a smooth nowhere vanishing top dimensional form on . An function on is BV if, for all , the distributional derivative exists and
Theorem 5 (Characterization) Let be on . Then the distributional derivative exists if and only iff
Theorem 6 (Structure) Let be in BV. Then
- The map, defined on open sets by
is the restriction of a measure.
- There is a Borel measurable vector field on such that - almost everywhere, and which achieves the supremum.
Note that a more general theory of BV functions on metric spaces has been started by Ambrosio and Miranda in the early 2000′s. Our class of BV functions is a priori larger than theirs (where the metric is the Carnot-Carathéodory metric associated to ). It coincides with theirs in the sub-Riemannian case (i.e. vectors of finite length form a smooth sub-bundle), to which I will stick from now on.
2. Sets of finite perimeter
Definition 7 A subset has finite perimeter if .
In the Euclidean case, De Giorgi’s rectifiability theorem states
Theorem 8 (De Giorgi) In , if has finite perimeter, then is concentrated on the reduced boundary which is -rectifiable.
The main tool in the proof is the
Theorem 9 (Blow up) In , if has finite perimeter, then at -almost every point (Lebesgue points of ), dilates of converge to the indicator of a half space.
Note that . By homogeneity of , the dilated indicators are bounded in BV. Some weak limit exists, it is of the form where has finite perimeter, and its normal is constante. This implies that is a halfspace.
2.1. Sub-Riemannian case
De Giorgi’s theorem has been extended two step 2 Carnot groups by Franchi, Serapioni and Serra-Cassano. One difficulty If
then for all brackets .
2.2. Blow up
We have been able to generalize their result to sub-Riemannian manifolds with varying geometry.
Theorem 10 (Mostow-Margulis, Bellaïche) In a sub-Riemannian manifold, there is a metric tangent cone at each point, it is isometric to a sub-Riemannian structure on the Lie group generated by the homogeneous degree components of the generating vector fields.
Theorem 11 Let have finite perimeter. Let be a point of the reduced boundary . If the nilpotent aproximation at is a 2-step Carnot group, then blow ups of at converge to vertical half-spaces.
The proof relies on the asymptotic doubling property (general for metric measure spaces satisfying Poincaré inequality).
2.3. Open questions
For higher step groups, Ambrosio-Kleiner-Le Donne prove convergence of a subsequence of dilates to a vertical half-space. They cannot prove rectifiability of the reduced boundary.
There is an example of a set with constant normal in Engel’s group which is not a vertical half-space. Nevertheless, its blow-ups are half-spaces at all but one point. Therefore I still think that there should be a blow-up, a half-space, almost everywhere.