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.