Hausdorff volume in non equiregular sub-Riemannian manifolds.

*(English)*Zbl 1325.53039The authors study the relationship between smooth volume and Hausdorff volume on non-equiregular sub-Riemannian manifolds.

Let \(M\) be a sub-Riemannian manifold of dimension \(n\) with horizontal distribution \(\mathcal{D} \subset TM\), not necessarily of constant rank, and sub-Riemannian metric \(g\). Assuming that \(\mathcal{D}\) satisfies the usual Lie bracket generating condition, then \(M\) can be equipped with the Carnot-Carathéodory distance \(d\). We may consider the Hausdorff dimension \(Q\) of the metric space \((M,d)\), which is typically strictly greater than \(n\), and the corresponding (spherical) Hausdorff volume \(\mathrm{vol}_H\) on \(M\). Assuming that \(M\) is orientable and thus admits a smooth, positively oriented, non-degenerate \(n\)-form \(\omega\), we can also consider the corresponding smooth volume measure \(\mu\) on \(M\). (The choice of \(\omega\) is not important in this paper.) This paper examines how the two measures \(\mu\) and \(\mathrm{vol}_H\) relate to each other.

The focus here is on sub-Riemannian manifolds which are not equiregular. For each \(p \in M\), we may consider the flag \((\mathcal{D}_p = \mathcal{D}^1_p, \mathcal{D}^2_p, \dots, \mathcal{D}^{r(p)}_p)\) of the horizontal distribution \(\mathcal{D}\), where \(\mathcal{D}^i_p\) is the subspace of \(T_p M\) spanned by horizontal vector fields (smooth sections of \(\mathcal{D}\)) and their iterated Lie brackets up to order \(i\). Let \(n_i(p) = \dim \mathcal{D}^i_p\). If the weight vector \((n_1(p), \dots, n_{r(p)}(p))\) is the same for every \(p\), then \(M\) is said to be equiregular. Roughly speaking, this says that the process of generating \(T_p M\) from the Lie brackets of horizontal vector fields looks similar at all points. In this case, the Hausdorff dimension \(Q\) is simply given by \(Q = \sum_{i=1}^r i(n_i - n_{i-1})\), and it is known that the measures \(\mu\) and \(\mathrm{vol}_H\) are mutually absolutely continuous, and moreover are commensurable, i.e., the Radon-Nikodym derivatives \(\frac{d\mathrm{vol}_H}{d \mu}\) and \(\frac{d\mu}{d\mathrm{vol}_H}\) are both locally essentially bounded [A. Agrachev et al., Calc. Var. Partial Differ. Equ. 43, No. 3–4, 355–388 (2012; Zbl 1236.53030); J. Mitchell, J. Differ. Geom. 21, 35–45 (1985; Zbl 0554.53023); R. Montgomery, A tour of sub-Riemannian geometries, their geodesics and applications. Mathematical Surveys and Monographs 91. Providence, RI: American Mathematical Society (AMS) (2002; Zbl 1044.53022)]. Although these results were previously known, they also follow as a special case of the results of this paper (Theorem 3.1). A particular consequence is that \(\mathrm{vol}_H\) is a Radon measure.

However, the emphasis of this paper is on the more subtle case that \(M\) is not equiregular. In this case we may partition \(M\) into regular and singular sets. A point \(p \in M\) is called regular if the weight vector \(q \mapsto (n_1(q), \dots, n_{r(q)}(q))\) is constant in a neighborhood of \(p\); otherwise \(p\) is called singular. (Roughly speaking, at a singular point \(p\), the process of generating \(T_p M\) from Lie brackets of horizontal vector field happens differently than at nearby points.) The sets of all regular and singular points are denoted \(\mathsf{R}\) and \(\mathsf{S}\) respectively. The authors work under the assumption that \(\mu(\mathsf{S})=0\), though they give an example where this does not hold (Example 3.4). The authors write \(\mathrm{vol}_H \llcorner_{\mathsf{R}}\) to denote the restriction of the measure \(\mathrm{vol}_H\) to the subset \(\mathsf{R} \subset M\) (this slightly uncommon notation is not defined in the paper, but the authors have confirmed its meaning in personal communication to the reviewer), and \(\mathrm{vol}_H \llcorner_{\mathsf{S}}\) is defined analogously. The first main result of this paper (Corollary 3.9) is that the Lebesgue decomposition of \(\mathrm{vol}_H\) with respect to \(\mu\) is given by \(\mathrm{vol}_H = \mathrm{vol}_H \llcorner_{\mathsf{R}} + \mathrm{vol}_H \llcorner_{\mathsf{S}}\), where \(\mathrm{vol}_H \llcorner_{\mathsf{R}} \ll \mu\) and \(\mathrm{vol}_H \llcorner_{\mathsf{S}} \perp \mu\). (Note that either of these measures could potentially be 0 if \(\mathsf{R}\) or \(\mathsf{S}\) has zero Hausdorff volume, in which case we would have \(\mathrm{vol}_H \perp \mu\) or \(\mathrm{vol}_H \ll \mu\).)

To get more detailed information about the relationship between \(\mathrm{vol}_H\) and \(\mu\), the authors work under the additional assumption that \(M\) is stratified by equisingular submanifolds. That is, the singular set \(\mathsf{S}\) should be able to be written as a locally finite stratification of submanifolds \(\mathsf{S}_i\), each of which is equisingular; this means, roughly, that the weight vector is constant on each \(\mathsf{S}_i\), whether considered with respect to the ambient manifold \(M\) or internally to the submanifold \(\mathsf{S}_i\).

Under this assumption, it turns out that the relationship between \(\mathrm{vol}_H\) and \(\mu\) depends on the relative Hausdorff dimensions \(Q_{\mathsf{R}}, Q_{\mathsf{S}}\) of \(\mathsf{R}, \mathsf{S}\) respectively. The results of this paper give a rather complete picture of this relationship, summarized as follows (taken from the paper’s Figure 1).

1. If \(\mathsf{S} = \emptyset\) (the equiregular case), then \(\mathrm{vol}_H\) is Radon and is mutually absolutely continuous and commensurable to \(\mu\).

2. If \(\mathsf{S} \neq \emptyset\) but \(Q_{\mathsf{S}} < Q_{\mathsf{R}}\), then \(\mathrm{vol}_H \ll \mu\) but the measures are not commensurable. \(\mathrm{vol}_H\) may or may not be Radon, depending on the particular way in which the flag varies around the singular set (Propositions 4.4, 4.9, 4.10).

3. If \(Q_{\mathsf{S}} = Q_{\mathsf{R}}\) then \(\mathrm{vol}_H\) is not Radon (Corollary 4.6), nor is it absolutely continuous to \(\mu\), and the absolutely continuous part \(\mathrm{vol}_H \llcorner_{\mathsf{R}}\) is not commensurable to \(\mu\).

4. If \(Q_{\mathsf{S}} > Q_{\mathsf{R}}\) then \(\mathrm{vol}_H\) and \(\mu\) are mutually singular.

The authors proceed by a detailed analysis of the relationship between Hausdorff volume and smooth volume within equisingular submanifolds, which may be of interest in its own right. A key tool is the ability to work within a system of privileged coordinates, and a version of the ball-box theorem for equisingular submanifolds (Proposition A.1). Several instructive examples are included, which help to illustrate the various different cases under consideration.

Let \(M\) be a sub-Riemannian manifold of dimension \(n\) with horizontal distribution \(\mathcal{D} \subset TM\), not necessarily of constant rank, and sub-Riemannian metric \(g\). Assuming that \(\mathcal{D}\) satisfies the usual Lie bracket generating condition, then \(M\) can be equipped with the Carnot-Carathéodory distance \(d\). We may consider the Hausdorff dimension \(Q\) of the metric space \((M,d)\), which is typically strictly greater than \(n\), and the corresponding (spherical) Hausdorff volume \(\mathrm{vol}_H\) on \(M\). Assuming that \(M\) is orientable and thus admits a smooth, positively oriented, non-degenerate \(n\)-form \(\omega\), we can also consider the corresponding smooth volume measure \(\mu\) on \(M\). (The choice of \(\omega\) is not important in this paper.) This paper examines how the two measures \(\mu\) and \(\mathrm{vol}_H\) relate to each other.

The focus here is on sub-Riemannian manifolds which are not equiregular. For each \(p \in M\), we may consider the flag \((\mathcal{D}_p = \mathcal{D}^1_p, \mathcal{D}^2_p, \dots, \mathcal{D}^{r(p)}_p)\) of the horizontal distribution \(\mathcal{D}\), where \(\mathcal{D}^i_p\) is the subspace of \(T_p M\) spanned by horizontal vector fields (smooth sections of \(\mathcal{D}\)) and their iterated Lie brackets up to order \(i\). Let \(n_i(p) = \dim \mathcal{D}^i_p\). If the weight vector \((n_1(p), \dots, n_{r(p)}(p))\) is the same for every \(p\), then \(M\) is said to be equiregular. Roughly speaking, this says that the process of generating \(T_p M\) from the Lie brackets of horizontal vector fields looks similar at all points. In this case, the Hausdorff dimension \(Q\) is simply given by \(Q = \sum_{i=1}^r i(n_i - n_{i-1})\), and it is known that the measures \(\mu\) and \(\mathrm{vol}_H\) are mutually absolutely continuous, and moreover are commensurable, i.e., the Radon-Nikodym derivatives \(\frac{d\mathrm{vol}_H}{d \mu}\) and \(\frac{d\mu}{d\mathrm{vol}_H}\) are both locally essentially bounded [A. Agrachev et al., Calc. Var. Partial Differ. Equ. 43, No. 3–4, 355–388 (2012; Zbl 1236.53030); J. Mitchell, J. Differ. Geom. 21, 35–45 (1985; Zbl 0554.53023); R. Montgomery, A tour of sub-Riemannian geometries, their geodesics and applications. Mathematical Surveys and Monographs 91. Providence, RI: American Mathematical Society (AMS) (2002; Zbl 1044.53022)]. Although these results were previously known, they also follow as a special case of the results of this paper (Theorem 3.1). A particular consequence is that \(\mathrm{vol}_H\) is a Radon measure.

However, the emphasis of this paper is on the more subtle case that \(M\) is not equiregular. In this case we may partition \(M\) into regular and singular sets. A point \(p \in M\) is called regular if the weight vector \(q \mapsto (n_1(q), \dots, n_{r(q)}(q))\) is constant in a neighborhood of \(p\); otherwise \(p\) is called singular. (Roughly speaking, at a singular point \(p\), the process of generating \(T_p M\) from Lie brackets of horizontal vector field happens differently than at nearby points.) The sets of all regular and singular points are denoted \(\mathsf{R}\) and \(\mathsf{S}\) respectively. The authors work under the assumption that \(\mu(\mathsf{S})=0\), though they give an example where this does not hold (Example 3.4). The authors write \(\mathrm{vol}_H \llcorner_{\mathsf{R}}\) to denote the restriction of the measure \(\mathrm{vol}_H\) to the subset \(\mathsf{R} \subset M\) (this slightly uncommon notation is not defined in the paper, but the authors have confirmed its meaning in personal communication to the reviewer), and \(\mathrm{vol}_H \llcorner_{\mathsf{S}}\) is defined analogously. The first main result of this paper (Corollary 3.9) is that the Lebesgue decomposition of \(\mathrm{vol}_H\) with respect to \(\mu\) is given by \(\mathrm{vol}_H = \mathrm{vol}_H \llcorner_{\mathsf{R}} + \mathrm{vol}_H \llcorner_{\mathsf{S}}\), where \(\mathrm{vol}_H \llcorner_{\mathsf{R}} \ll \mu\) and \(\mathrm{vol}_H \llcorner_{\mathsf{S}} \perp \mu\). (Note that either of these measures could potentially be 0 if \(\mathsf{R}\) or \(\mathsf{S}\) has zero Hausdorff volume, in which case we would have \(\mathrm{vol}_H \perp \mu\) or \(\mathrm{vol}_H \ll \mu\).)

To get more detailed information about the relationship between \(\mathrm{vol}_H\) and \(\mu\), the authors work under the additional assumption that \(M\) is stratified by equisingular submanifolds. That is, the singular set \(\mathsf{S}\) should be able to be written as a locally finite stratification of submanifolds \(\mathsf{S}_i\), each of which is equisingular; this means, roughly, that the weight vector is constant on each \(\mathsf{S}_i\), whether considered with respect to the ambient manifold \(M\) or internally to the submanifold \(\mathsf{S}_i\).

Under this assumption, it turns out that the relationship between \(\mathrm{vol}_H\) and \(\mu\) depends on the relative Hausdorff dimensions \(Q_{\mathsf{R}}, Q_{\mathsf{S}}\) of \(\mathsf{R}, \mathsf{S}\) respectively. The results of this paper give a rather complete picture of this relationship, summarized as follows (taken from the paper’s Figure 1).

1. If \(\mathsf{S} = \emptyset\) (the equiregular case), then \(\mathrm{vol}_H\) is Radon and is mutually absolutely continuous and commensurable to \(\mu\).

2. If \(\mathsf{S} \neq \emptyset\) but \(Q_{\mathsf{S}} < Q_{\mathsf{R}}\), then \(\mathrm{vol}_H \ll \mu\) but the measures are not commensurable. \(\mathrm{vol}_H\) may or may not be Radon, depending on the particular way in which the flag varies around the singular set (Propositions 4.4, 4.9, 4.10).

3. If \(Q_{\mathsf{S}} = Q_{\mathsf{R}}\) then \(\mathrm{vol}_H\) is not Radon (Corollary 4.6), nor is it absolutely continuous to \(\mu\), and the absolutely continuous part \(\mathrm{vol}_H \llcorner_{\mathsf{R}}\) is not commensurable to \(\mu\).

4. If \(Q_{\mathsf{S}} > Q_{\mathsf{R}}\) then \(\mathrm{vol}_H\) and \(\mu\) are mutually singular.

The authors proceed by a detailed analysis of the relationship between Hausdorff volume and smooth volume within equisingular submanifolds, which may be of interest in its own right. A key tool is the ability to work within a system of privileged coordinates, and a version of the ball-box theorem for equisingular submanifolds (Proposition A.1). Several instructive examples are included, which help to illustrate the various different cases under consideration.

Reviewer: rough

##### MSC:

53C17 | Sub-Riemannian geometry |

28A78 | Hausdorff and packing measures |

28A80 | Fractals |

58C35 | Integration on manifolds; measures on manifolds |

28C15 | Set functions and measures on topological spaces (regularity of measures, etc.) |

28A75 | Length, area, volume, other geometric measure theory |

49Q15 | Geometric measure and integration theory, integral and normal currents in optimization |

##### Keywords:

sub-Riemannian geometry; Hausdorff measures; intrinsic volumes; geometric measure theory; equisingular submanifolds
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\textit{R. Ghezzi} and \textit{F. Jean}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 126, 345--377 (2015; Zbl 1325.53039)

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