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Register a new userHorizontally affine functions on step-2 Carnot algebras
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In this paper we introduce the notion of horizontally affine, h-affine in short, function and give a complete description of such functions on step-2 Carnot algebras. We show that the vector space of h-affine functions on the free step-2 rank-$n$ Carnot algebra is isomorphic to the exterior algebra of $mathbb{R}^n$. Using that every Carnot algebra can be written as a quotient of a free Carnot algebra, we shall deduce from the free case a description of h-affine functions on arbitrary step-2 Carnot algebras, together with several characterizations of those step-2 Carnot algebras where h-affine functions are affine in the usual sense of vector spaces. Our interest for h-affine functions stems from their relationship with a class of sets called precisely monotone, recently introduced in the literature, as well as from their relationship with minimal hypersurfaces.
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In this paper, we construct Holder maps to Carnot groups equipped with a Carnot metric, especially the first Heisenberg group $mathbb{H}$. Pansu and Gromov observed that any surface embedded in $mathbb{H}$ has Hausdorff dimension at least 3, so there is no $alpha$-Holder embedding of a surface into $mathbb{H}$ when $alpha>frac{2}{3}$. Zust improved this result to show that when $alpha>frac{2}{3}$, any $alpha$-Holder map from a simply-connected Riemannian manifold to $mathbb{H}$ factors through a metric tree. In the present paper, we show that Zusts result is sharp by constructing $(frac{2}{3}-epsilon)$-Holder maps from $D^2$ and $D^3$ to $mathbb{H}$ that do not factor through a tree. We use these to show that if $0<alpha < frac{2}{3}$, then the set of $alpha$-Holder maps from a compact metric space to $mathbb{H}$ is dense in the set of continuous maps and to construct proper degree-1 maps from $mathbb{R}^3$ to $mathbb{H}$ with Holder exponents arbitrarily close to $frac{2}{3}$.
We show that every finite-dimensional Alexandrov space X with curvature bounded from below embeds canonically into a product of an Alexandrov space with the same curvature bound and a Euclidean space such that each affine function on X comes from an affine function on the Euclidean space.
We prove that in arbitrary Carnot groups $mathbb G$ of step 2, with a splitting $mathbb G=mathbb Wcdotmathbb L$ with $mathbb L$ one-dimensional, the graph of a continuous function $varphicolon Usubseteq mathbb Wto mathbb L$ is $C^1_{mathrm{H}}$-regular precisely when $varphi$ satisfies, in the distributional sense, a Burgers type system $D^{varphi}varphi=omega$, with a continuous $omega$. We stress that this equivalence does not hold already in the easiest step-3 Carnot group, namely the Engel group. As a tool for the proof we show that a continuous distributional solution $varphi$ to a Burgers type system $D^{varphi}varphi=omega$, with $omega$ continuous, is actually a broad solution to $D^{varphi}varphi=omega$. As a by-product of independent interest we obtain that all the continuous distributional solutions to $D^{varphi}varphi=omega$, with $omega$ continuous, enjoy $1/2$-little Holder regularity along vertical directions.
The Whitney extension theorem is a classical result in analysis giving a necessary and sufficient condition for a function defined on a closed set to be extendable to the whole space with a given class of regularity. It has been adapted to several settings, among which the one of Carnot groups. However, the target space has generally been assumed to be equal to R^d for some d $ge$ 1. We focus here on the extendability problem for general ordered pairs (G_1,G_2) (with G_2 non-Abelian). We analyze in particular the case G_1 = R and characterize the groups G_2 for which the Whitney extension property holds, in terms of a newly introduced notion that we call pliability. Pliability happens to be related to rigidity as defined by Bryant an Hsu. We exploit this relation in order to provide examples of non-pliable Carnot groups, that is, Carnot groups so that the Whitney extension property does not hold. We use geometric control theory results on the accessibility of control affine systems in order to test the pliability of a Carnot group. In particular, we recover some recent results by Le Donne, Speight and Zimmermann about Lusin approximation in Carnot groups of step 2 and Whitney extension in Heisenberg groups. We extend such results to all pliable Carnot groups, and we show that the latter may be of arbitrarily large step.
We provide a Rademacher theorem for intrinsically Lipschitz functions $phi:Usubseteq mathbb Wto mathbb L$, where $U$ is a Borel set, $mathbb W$ and $mathbb L$ are complementary subgroups of a Carnot group, where we require that $mathbb L$ is a normal subgroup. Our hypotheses are satisfied for example when $mathbb W$ is a horizontal subgroup. Moreover, we provide an area formula for this class of intrinsically Lipschitz functions.
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