Notes of Roberta Ghezzi’s lecture

BV functions and sets of finite perimeter in SR manifolds

Roberta Ghezzi

with Luigi Ambrosio and Valentino Magnani

1. BV functions on manifolds

Definition 1 Let {\Omega} be an open set in {{\mathbb R}^n}. An {L^1} function {u} on {M} is BV if

\displaystyle  \begin{array}{rcl}  \sup\{\int_{\Omega}u\,div(X)\,dx\,;\,X\,C^1\textrm{compactly supported vector field on }\Omega,\,\|X\|_{\infty}\leq 1\}<\infty. \end{array}

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 {M} be a smooth oriented manifold, {\omega} a smooth nowhere vanishing top dimensional form on {M}, {X} a {C^1} compactly supported vector field on {M}. The divergence of {X} is the function such that

\displaystyle  \begin{array}{rcl}  L_X \omega=div(X)\omega. \end{array}

Definition 3 Let {G:TM\rightarrow[0,+\infty]} be a smooth function which is quadratic on fibers. Let {\Gamma^G} denote the class of {C^1} vector fields on {M} such that {G(X)\leq 1} everywhere.

Definition 4 Let {M} be a smooth oriented manifold, {\omega} a smooth nowhere vanishing top dimensional form on {M}. An {L^1} function {u} on {M} is BV if, for all {X\in\Gamma^G}, the distributional derivative {D_X u} exists and

\displaystyle  \begin{array}{rcl}  \sup_{X\in\Gamma^G}|D_X u|(\Omega) <\infty. \end{array}

Theorem 5 (Characterization) Let {u} be {L^1} on {M}. Then the distributional derivative {Xu} exists if and only iff

\displaystyle  \begin{array}{rcl}  \sup_{K\subset\subset M}\{\lim_{t\rightarrow 0}\int_{K}\frac{|u(\phi_t^X(q)-u(q)|}{|t|}\omega\}<\infty. \end{array}

Theorem 6 (Structure) Let {u} be in BV. Then

  1. The map, defined on open sets {A} by

    \displaystyle  \begin{array}{rcl}  \|D_G u\|(A):=\sup_{X\in\Gamma^G}|D_X u|(A) \end{array}

    is the restriction of a measure.

  2. There is a Borel measurable vector field {\nu_u} on {M} such that {G(\nu_u)=1}{\|D_G u\|} 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 {G}). 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 {E\subset M} has finite perimeter if {1_E \in BV}.

In the Euclidean case, De Giorgi’s rectifiability theorem states

Theorem 8 (De Giorgi) In {{\mathbb R}^n}, if {E} has finite perimeter, then {D_G 1_E} is concentrated on the reduced boundary {\mathcal{F}^* E} which is {(n-1)}-rectifiable.

The main tool in the proof is the

Theorem 9 (Blow up) In {{\mathbb R}^n}, if {E} has finite perimeter, then at {D_G 1_E}-almost every point (Lebesgue points of {\nu_E}), dilates of {E} converge to the indicator of a half space.

Note that {(D_{X_1}1_E,\ldots,D_{X_n}1_E)=\nu_E \|D_G 1_E\|}. By homogeneity of { \|D_G 1_E\|}, the dilated indicators are bounded in BV. Some weak limit exists, it is of the form {1_F} where {F} has finite perimeter, and its normal {\nu_F} is constante. This implies that {F} 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

\displaystyle  \begin{array}{rcl}  D_{X_1}1_E\geq 0,\quad D_{X_2}1_E =0,...,\quad D_{X_m}1_E=0. \end{array}

then {D_{Y}1_E=0} for all brackets {Y=[X_1,X_j]}.

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 {-1} components of the generating vector fields.

Theorem 11 Let {E} have finite perimeter. Let {p} be a point of the reduced boundary {\mathcal{F}^* E}. If the nilpotent aproximation at {p} is a 2-step Carnot group, then blow ups of {E} at {p} 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.

About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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