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On singular sets of \(H\)-monotone maps. (English) Zbl 1431.53030

Let \(\mathbb{H}^{n}\subset \mathbb{R}^{2n+1}\) be the Heisenberg group and \(T:\mathbb{H}^{n}\rightrightarrows \mathbb{R}^{2n}\) be maximal (with respect to graph inclusion) H-monotone (in the sense of A. Calogero and R. Pini [J. Convex Anal. 19, No. 2, 541–567 (2012; Zbl 1247.52002)]), and assume that it takes nonempty values everywhere. For \(p\in \mathbb{H}^{n}\), let \(V_{p}\) be the vector space generating the smallest affine manifold containing \(p,\) and denote its dimension by \(k\). Define \(\Sigma _{H}^{k}(T)=\{p\in \mathbb{H} ^{n}\mid V_{p}\in G^{H}(2n,k)\}\); here \(G^{H}(2n,k)\) stands for the Grassmanian of horizontal \(k\)-dimensional subspaces of \(\mathbb{R}^{2n}\), considered by the first author et al. [Adv. Math. 231, No. 2, 569–604 (2012; Zbl 1260.28007)]. The main result in this paper establishes that, if \(k\in \{1,\dots,n\}\), then the set \(\Sigma _{H}^{k}(T)\) is \((2n+1-k,\mathbb{H})\)-rectifiable in the following sense: There exists a sequence of \((2n+1-k)\)-dimensional \(\mathbb{H}\)-regular surfaces \((S_{i})_{i\in \mathbb{N}}\) such that \(\mathcal{S}^{2n+2-k}(E\setminus \bigcup\limits_{i\in \mathbb{N}}S_{i})=0\); here \( \mathcal{S}^{2n+2-k}\) denotes the \((2n+2-k)\)-dimensional spherical Hausdorff measure with respect to the Korányi metric \(d\), defined by \( d(p,q):=\left\Vert p^{-1}q\right\Vert ,\) with \(\left\Vert \cdot \right\Vert \) denoting the Korányi norm on \(\mathbb{H}^{n}\), given by \(\left\Vert (x,y,t)\right\Vert :=\sqrt[4]{\left\Vert (x,y)\right\Vert _{\mathbb{R} ^{2n}}^{4}+t^{2}}\), and \((x,y,t)^{-1}:=(-x,-y,-t)\).

MSC:

53C17 Sub-Riemannian geometry
47H05 Monotone operators and generalizations
49J53 Set-valued and variational analysis
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References:

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