Regularity properties of spheres in homogeneous groups

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC-BY-NC-ND 4.0 https://creativecommons.org/licenses/by-nc-nd/4.0/ Regularity properties of spheres in homogeneous groups © 2017 American Mathematical Society. Accepted version (Final draft) Le Donne, Enrico; Nicolussi Golo, Sebastiano Le Donne, E., & Nicolussi Golo, S. (2018). Regularity properties of spheres in homogeneous groups. Transactions of the American Mathematical Society, 370, 2057-2084. https://doi.org/10.1090/tran/7038 2018 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000–000 S 0002-9947(XX)0000-0 REGULARITY PROPERTIES OF SPHERES IN HOMOGENEOUS GROUPS ENRICO LE DONNE AND SEBASTIANO NICOLUSSI GOLO Abstract. We study left-invariant distances on Lie groups for which there exists a one-parameter family of homothetic automorphisms. The main examples are Carnot groups, in particular the Heisenberg group with the standard dilations. We are interested in criteria implying that, locally and away from the diagonal, the distance is Euclidean Lipschitz and, consequently, that the metric spheres are boundaries of Lipschitz domains in the Euclidean sense. In the first part of the paper, we consider geodesic distances. In this case, we actually prove the regularity of the distance in the more general context of sub- Finsler manifolds with no abnormal geodesics. Secondly, for general groups we identify an algebraic criterium in terms of the dilating automorphisms, which for example makes us conclude the regularity of every homogeneous distance on the Heisenberg group. In such a group, we analyze in more details the geometry of metric spheres. We also provide examples of homogeneous groups where spheres present cusps. Contents 1. Introduction 2 2. Preliminaries 4 2.1. Sub-Finsler structures 4 2.2. Graded groups 6 3. Regularity of sub-Finsler distances 8 3.1. About the minimal stretching 10 3.2. The End-point map is weakly* continuous 10 3.3. The differential of the End-point map is an End-point map 12 3.4. The sub-Finsler distance is Lipschitz in absence of singular geodesics 13 4. Regularity of spheres in graded groups 14 4.1. The sphere as a graph 14 4.2. An intrinsic approach 15 5. Examples 17 5.1. Three gradings on R217 5.2. The Heisenberg groups 21 5.3. A sub-Finsler sphere with a cusp 21 5.4. A sub-Riemannian sphere with a cusp 21 Received by the editors July 22, 2016. 2010 Mathematics Subject Classification. 28A75, 22E25, 53C60, 53C17, 26A16. E.L.D. has been supported by the Academy of Finland project no. 288501. S.N.G. has been supported by the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme FP7/2007-2013/ under REA grant agreement n. 607643. c XXXX American Mathematical Society 1 2 LE DONNE AND NICOLUSSI GOLO 6. A closer look at the Heisenberg group 22 6.1. Proof of Proposition 6.2 23 6.2. Proof of Proposition 6.3 24 Appendix A. Equivalence of some definitions and existence of singular minimizers 26 Acknowledgment 27 References 28 1. Introduction The study of the asymptotic geometry of groups lead us to investigate spheres in homogeneous groups, examples of which are asymptotic cones of finitely generated nilpotent groups. A homogeneous group is a Lie group Gendowed with a family of Lie group automorphisms {δλ}λ>0and a left-invariant distance dfor which each δλmultiplies the distance by λ, see Section 2.2. An algebraic characterization of these groups is known by [29]. In fact, the Lie algebra gof Gadmits a grading, i.e., a decomposition g=Li≥1Visuch that [Vi, Vj]⊂Vi+j. For simplicity, we assume that the dilations are the ones induced by the grading. Namely, the dilation of factor λrelative to the grading is the one such that (δλ)∗(v) = λivfor all v∈Vi. We denote by 0 the neutral element of Gand by Sdthe unit sphere at 0 for a distance don G, i.e., Sd:= {p∈G:d(0, p) = 1}. In this paper we want to exclude cusps in spheres since their presence in the asymptotic cone of a finitely generated nilpotent group may give a slower rate of convergence in the blow down, see [8]. We find criteria implying that the metric spheres are boundaries of Lipschitz domains and in fact that the distance function from a point is a locally Lipschitz function with respect to a Riemannian metric. First, we address the case where the distance dis a length distance. Thanks to a characterization of Carnot groups, see [18], the group Gis in this case a stratified group and dis a sub-Finsler distance. Being a stratified group means that the grading of gis such that the first layer V1generates g. Being a sub-Finsler distance means that there are a left-invariant subbundle ∆ ⊂TG and a left-invariant norm k · k on ∆ such that the length induced by dof an absolutely continuous curve γ: [0,1] →Gis equal to R1 0kγ0(t)kdt, where kγ0(t)k= +∞if γ0(t)/∈∆. The left-invariant subbundle ∆ is in fact the one generated by V1. In the sub-Finsler case, an obstruction to Lipschitz regularity of the sphere comes from the presence of length-minimizing curves (also called geodesics) that are not regular, in the sense that the first variation parallel to the subbundle ∆ does not have maximal rank, see Definition 2.6. Theorem 1.1. Let Gbe a stratified group endowed with a sub-Finsler metric d. Let d0:G→[0,+∞),p7→ d(0, p). Let p∈Gbe such that all geodesics from 0to p are regular. Then for any Riemannian metric ρon Gthe function d0is Lipschitz with respect to ρin some neighborhood of p. We will actually state and prove Theorem 1.1 in the more general setting of sub-Finsler manifolds of constant-type norm, see Section 2.1. REGULARITY OF SPHERES 3 In case of homogeneity, the regularity of the distance implies also the regularity of the spheres. Hence, using Theorem 1.1 we easily get the second result for sub- Finsler homogeneous groups. Theorem 1.2. Let Gbe a stratified group endowed with a sub-Finsler metric d. Let p∈Sdbe such that all geodesics from 0to pare regular. Then, in smooth coordinates, the set Sdis a Lipschitz graph in some neighborhood of p. In particular, if all non-constant geodesics are regular, then metric balls are Lipschitz domains. Notice that a ball may be a Lipschitz domain even if the distance from a point is not Lipschitz (we give an example in Remark 5.5). In Section 5 we also present examples of sub-Riemannian and sub-Finsler distances whose balls have a cusp. At a second stage, we drop the hypothesis of dbeing a length distance and we present a result similar to the previous Theorem 1.2 in the context of homogeneous groups. Hereafter we denote by Lpand Rpthe left and the right translations on G, respectively, and by ¯ δ(p) the vector d dtδt(p)t=1 ∈TpG, where {δt}t>0are the dilations relative to a grading. Theorem 1.3. Let (G, d)be a homogeneous group with dilations relative to a grading, see Definition 2.12. Assume p∈Sdis such that (1.1) dLp(V1) + dRp(V1) + span{¯ δ(p)}=TpG. Then, in some neighborhood of pwe have that the sphere Sdis a Lipschitz graph and the distance d0from the identity is Lipschitz with respect to any Riemannian metric ρ. The similarity between Theorem 1.2 and Theorem 1.3 consists in the fact that, if dis a sub-Finsler distance, then condition (1.1) implies that all geodesics from 0 to pare regular, see Remark 4.5. The equality (1.1) or the absence of non-regular geodesics are actually quite strong conditions. However, in general we can give an upper bound for the Hausdorff dimension of spheres. In fact, if dis a homogeneous distance on a graded group of maximal degree s, then (1.2) dimρ H(G)−1≤dimρ H(Sd)≤dimρ H(G)−1 s, where dimρ His the Hausdorff dimension with respect to some (therefore any) Riemannian metric ρ. We show with Proposition 5.1 that this estimate is sharp. In the last part of the paper, we analyze in more details an important specific example: the Heisenberg group. In this graded group we consider all possible homogeneous distances and prove that in exponential coordinates (i) the unit ball is a star-shaped Lipschitz domain (Proposition 6.1); (ii) the unit sphere is a locally Lipschitz graph with respect to the direction of the center of the group (Proposition 6.2). We also give a method to construct homogeneous distances in the Heisenberg group with arbitrary Lipschitz regularity of the sphere. Namely, the graph of each Lipschitz function defined on the unit disk, up to adding to it a constant, is the sphere of some homogeneous distance, see Proposition 6.3. The investigation of this class of 4 LE DONNE AND NICOLUSSI GOLO examples is meaningful in connection to Besicovitch’s covering property as studied in [21] and [22]. The paper is organized as follows. In section 2 we will present all preliminary notions needed in the paper. We introduce sub-Finsler manifolds of constant-type norm, graded and homogeneous groups and Carnot groups. Section 3 is devoted to the proof of Theorem 1.1; first in the setting of sub-Finsler manifolds, see Theorem 3.1 proved in Section 3.4, then with a more specific result for Carnot groups, see Proposition 3.3. In Section 4 we see metric spheres as graphs over smooth spheres. Hence, we show Theorem 1.2, the inequalities (1.2), and Theorem 1.3. In Section 5 we present six examples: three different grading of R2, the Heisenberg group, a sub-Finsler sphere with a cusp and a sub-Riemannian sphere with a cusp. In Section 6 we prove stronger properties for spheres of homogeneous distances on the Heisenberg group. 2. Preliminaries 2.1. Sub-Finsler structures. Let Mbe a manifold of dimension n. We will write TM for the tangent bundle and Vec(M) for the space of smooth vector fields on M. Definition 2.1 (Sub-Finsler structure).Asub-Finsler structure (of constant-type norm) of rank ron a manifold Mis a triple (E,k·k,f), where (E,k·k) is a normed vector space of dimension rand f:M×E→TM is a smooth bundle morphism with f({p}×E)⊂TpM, for all p∈M. We added the specification “of constant-type norm” because the norm k·kdefined on the fibers of M×Edoes not depend on the base point of each fiber. Definition 2.2 (Horizontal curve).A curve γ: [0,1] →Mis a horizontal curve if it is absolutely continuous and there is u: [0,1] →Emeasurable, which is called a control of γ, such that γ0(t) = f(γ(t), u(t)) for a.e. t∈[0,1]. In this case γis called integral curve of uand we write γu. Definition 2.3 (Space of controls).The space of L∞-controls is defined as1 L∞([0,1]; E) := (u: [0,1] →Emeasurable, ess sup t∈[0,1] ku(t)k<∞). This is a Banach space with norm kukL∞:= ess supt∈[0,1] ku(t)k. Thanks to known results for ordinary differential equations, see [27], given a control u∈L∞([0,1]; E) and a point p∈Mthere is a unique solution γu,p to the Cauchy problem (γu,p(0) = p γ0 u,p(t) = f(γu,p(t), u(t)) for a.e. tin a neighborhood of 0. 1Among the three norms L1,L2and L∞for controls, we chose the latter because the unit ball in L1([0,1]; E) is not weakly compact and the L2-space is not a Hilbert space in our context. REGULARITY OF SPHERES 5 Remark 2.4.We will always assume that every u∈L∞([0,1]; E) and every p∈M the curve γu,p is defined on the interval [0,1]. This happens in many cases, for example for left-invariant sub-Finsler structures on Lie groups, in particular in Carnot groups. Definition 2.5 (End-point map).Fix o∈M. Define the End-point map with base point o,Endo:L∞([0,1]; E)→M, as Endo(u) = γu,o(1). By standard result of ODE the map Endois of class C1, see [27]. Definition 2.6 (Regular curves).Given o∈M, a control u∈L∞([0,1]; E) is said to be regular if it is a regular point of Endo, i.e., if dEndo(u) : L∞([0,1]; E)→TEndo(u)M is surjective. A singular control is a control that is not regular. Definition 2.7 (Sub-Finsler distance).The sub-Finsler distance, also called Carnot- Carath´eodory distance, between two points p, q ∈Mis d(p, q) := inf Z1 0ku(t)kdt:u∈L∞([0,1]; E) with Endp(u) = q. Clearly (M, d) is a metric space, even though it might happen d(p, q) = ∞. Let `d(γ) be the length of a curve γwith respect to d, see [4]. It can be proven that a curve γ: [0,1] →(M, d) is Lipschitz if and only if it is horizontal and it admits a control in L∞([0,1]; E). Moreover, if γis Lipschitz, then `d(γ) = inf Z1 0ku(t)kdt:u∈L∞([0,1]; E) control of γ. We will use the term geodesic as a synonym of length-minimizer. The distance can be expressed by using the L∞-norm, i.e., for every p, q ∈M d(p, q) = inf {kukL∞:u∈L∞([0,1]; E) with Endp(u) = q}. Moreover, if urealizes the infimum above, then its integral curve γustarting from pis a length-minimizing curve parametrized by constant velocity, i.e., d(p, q) = kukL∞=`d(γu) = ku(t)k,for a.e.t∈[0,1]. Notice that the L∞-norm plays a similar role here as the L2-energy in sub-Riemannian geometry. Definition 2.8 (Bracket-generating condition).Let Abe the Lie algebra generated by the set {p7→ f(p, X(p)) with X:M→Esmooth} ⊂ Vec(M). We say that the sub-Finsler structure (E,k·k,f) on Msatisfies the bracket-generating condition if for all p∈M {V(p) : V∈A}=TpM. As a consequence of the Orbit Theorem [17], we have the following basic wellknown fact. Lemma 2.9. If (E,k·k,f)satisfies the bracket-generating condition, then the distance dinduces the original topology of Mand (M, d)is a locally compact and locally geodesic length space. By the Hopf-Rinow Theorem, see [9], the assumption in Remark 2.4 implies that (M, d) is a complete, boundedly compact metric space. 6 LE DONNE AND NICOLUSSI GOLO 2.2. Graded groups. All Lie algebras considered here are over Rand finitedimensional. Definition 2.10 (Graded group).A Lie algebra gis graded if it is equipped with agrading, i.e., with a vector-space decomposition g=Li>0Vi, where i > 0 means i∈(0,∞), such that for all i, j > 0 it holds [Vi, Vj]⊂Vi+j. A graded Lie group is a simply connected Lie group Gwhose Lie algebra is graded. The maximal degree of a graded group Gis the maximum isuch that Vi6={0}. Graded groups are nilpotent and the exponential map exp : g→Gis a global diffeomorphism. We will denote by 0 the neutral element of Gand identify g=T0G. Definition 2.11 (Dilations).In a graded group for which the Lie algebra has the grading g=Li>0Vi, the dilations relative to the grading are the group homomorphisms δλ:G→G, for λ∈(0,∞), such that (δλ)∗(v) = λivfor all v∈Vi. In the definition above, φ∗denotes the Lie algebra homomorphism associated to a Lie group homomorphism φ, in particular, φ◦exp = exp ◦φ∗. Since a graded group is simply connected, δλis well defined. Notice that, for any λ, µ > 0, δλ◦δµ=δλµ. Definition 2.12 (Homogeneous distances).Let Gbe a graded group with a dilations {δλ}λ>0, relative to the grading. We say that a distance don Gis homogeneous if it is left-invariant, i.e., for every g, x, y ∈Gwe have d(gx, gy) = d(x, y), and onehomogeneous with respect to the dilations, i.e., for all λ > 0 and all x, y ∈G we have d(δλx, δλy) = λd(x, y). If dis one such a distance, then (G, d) is called homogeneous group (with dilations relative to the grading). Remark 2.13.A graded group admits a homogeneous distance if and only if for i∈(0,1) we have Vi={0}, see [16]. Given a homogeneous distance d, the function p7→ d0(p) := d(0, p) is a homogeneous norm. Here with the term homogeneous norm we mean a function N:G→[0,+∞) such that for all p, q ∈Gand all λ > 0 it holds (1) N(p)=0 ⇔p= 0; (2) N(pq)≤N(p) + N(q); (3) N(p−1) = N(p); (4) N(δλp) = λN(p). In fact, homogeneous distances are in bijection with homogeneous norms on G through the formula d(p, q) = N(p−1q). Homogeneous distances induce the original topology of G, see [22]. Moreover, given two homogeneous distances d1, d2on G, there is a constant C > 0 such that for all p, q ∈G (2.1) 1 Cd1(p, q)≤d2(p, q)≤Cd1(p, q). Lemma 2.14. Let Gbe a graded group and 0< k1≤k2such that Vi={0}for all i < k1and all i > k2. Let dbe a homogeneous distance and ρa left-invariant Riemannian metric on G. Then there are C,  > 0such that for all p, q ∈Gwith ρ(p, q)<  it holds (2.2) 1 Cρ(p, q)1 k1≤d(p, q)≤Cρ(p, q)1 k2. In particular, the homogeneous norm d0is locally 1 k2-H¨older. REGULARITY OF SPHERES 7 Proof. We identify G=gvia the exponential map. So, if p∈G, we denote by pi the i-th component in the decomposition p=Pipiwith pi∈Vi. Fix a norm |·| on g. For any pair (p, q)∈G×Gdefine η(p, q) := η(0, p−1q),where η(0, p) := max i(|pi|)1 i. The function ηis a so-called quasi-distance, see [22]. In particular, ηis continuous, left-invariant and one-homogeneous with respect to the dilations δλ. Therefore, if dis a homogeneous distance, then there is C > 0 such that 1 Cη(p, q)≤d(p, q)≤Cη(p, q). So, we can prove (2.2) only for η. Let C,  > 0 be with C < 1 and such that, if ρ(0, p)< , then (2.3) 1 Cρ(0, p)≤max i|pi| ≤ Cρ(0, p). Therefore, if ρ(p, q)< , then |(p−1q)i| ≤ Cρ(p, q)<1 for all iand (2.4) max i|(p−1q)i|1 k1≤max i(|(p−1q)i|)1 i=η(p, q)≤max i|(p−1q)i|1 k2, thanks to the monotonicity of the function x7→ axfor 0 <a<1. The thesis follows immediately from (2.3) and (2.4) combined.  Next lemma gives a characterization of sets that are the unit ball of a homogeneous distance. In this paper, we denote by int(B) the interior of a subset B. Lemma 2.15. Let Gbe a graded group with dilations δλ,λ > 0. A set B⊂G is the unit ball with center 0of a homogeneous distance on Gif and only if Bis compact, 0∈int(B),B=B−1and (2.5) ∀p, q ∈B, ∀t∈[0,1] δt(p)δ1−t(q)∈B. The proof of the latter fact is straightforward and hence omitted. One only needs to show that the function N(p) := inf{t≥0 : δt−1p∈B}is a homogeneous norm and B={p:N(p)≤1}. Definition 2.16 (Stratified group).Astratified group is a graded group Gsuch that its Lie algebra gis generated by the layer V1of the grading of g. Notice that in a stratified group Gthe maximal degree sof the grading equals the nilpotency step of Gand it holds g=Ls i=1 Viwith [V1, Vi] = Vi+1 for all i∈ {1, . . . , s}, with Vs+1 ={0}. We also remark that all stratifications of a group Gare isomorphic to each other, i.e., if g=Ls0 i=1 Wiis a second stratification, then there is a Lie group automorphism φ:G→Gsuch that φ∗(Wi) = Vifor all i, see [19]. In a stratified group, the map f:G×V1→TG,f(g, v) := dLg(v), is a bundle morphism with f(g, v)∈TgG. So, if k·kis any norm on V1, the triple (V1,k · k,f) is a sub-Finsler structure on G. The stratified group Gendowed with the corresponding sub-Finsler distance dis called Carnot group. Such a dis an example of a homogeneous distance on G. Remark 2.17.As already stated, singular curves play a central role in our analysis, because they disrupt the Lipschitz regularity of the distance function. We recall that every Carnot group of nilpotency step s≥3 has singular geodesics, see Appendix A. 8 LE DONNE AND NICOLUSSI GOLO More precisely, there is X∈V1such that the curve t7→ exp(tX) is a singular geodesic. In particular, if all non-constant length-minimizing curves are regular, then the step of the group is necessarily at most 2. 3. Regularity of sub-Finsler distances We will prove in this section that sub-Finsler distances are Lipschitz whenever all length-minimizing curves are regular, see Theorem 3.1. Theorem 1.1 expresses this result for Carnot groups. It is important to remind what is known in the sub-Riemannian case. A sub- Riemannian distance is a sub-Finsler distance whose norm on the bundle Eis induced by a scalar product. Rifford proved in [26] that, if there are no singular length-minimizers, for all o∈M, not only dois locally Lipschitz, but also the spheres centered at oare Lipschitz hypersurfaces for almost all radii. The key points of his proof are the tools of Clarke’s non-smooth calculus (see [12]) and a version of Sard’s Lemma for the distance function (see [25]). An exhaustive exposition of this topic can be found in [2]. In Rifford’s version of Sard’s Lemma, one uses the fact that the L2norm in the Hilbert space L2([0,1]; E) is smooth away from the origin. If Eis equipped with a generic norm, instead, the Lpnorm on Lp([0,1]; E) with 1 ≤p≤ ∞ may be non-smooth, hence the proof does not work in the sub-Finsler case. The non-smoothness of the norm can be seen in another dissimilarity between sub-Riemannian and sub-Finsler distances. Sub-Riemannian distances are proven to be locally semi-concave when there are no singular length-minimizing curves. We remind that a function f:Rn→Ris semi-concave if for each p∈Rnthere exists a C2function g:Rn→Rsuch that f≤gand f(p) = g(p), see [27]. Semi-concavity is a stronger property than being Lipschitz. However, semi-concavity fails to hold in the sub-Finsler case. For example, the `1-distance d(0,(x, y)) := |x|+|y|on R2is a sub-Finsler distance that is not semi-concave along the coordinate axis, although all curves are regular.2 We restrict our analysis to the Lipschitz regularity of the distance function, from which we deduce regularity properties of the spheres by means of the homogeneity of Carnot groups. With this aim in view, the core of the proof of Theorem 3.1 is the bound on the point-wise Lipschitz constant (see (3.5) at page 14), which already appeared in the sub-Riemannian context, see [1]. Our approach differs from the sub-Riemannian case for the fact that the set of optimal curves joining two points on a sub-Finsler manifold may not be compact in the W1,∞topology. As an example, consider the set of all length-minimizers from (0,0) to (0,1) for the `∞-distance d(0,(x, y)) := max{|x|,|y|} on R2.3However, we are still able to obtain a bound on the pointwise Lipschitz constant, i.e., to prove (3.5), by use of the weak* topology on controls. 2We show that d:R2×R2→Ris not locally semi-concave at the point ((0,0),(1,0)). Suppose there is a function φ∈C2(R2×R2) with φ((0,0),(1,0)) = d((0,0),(1,0)) = 1 and φ((x, y),(¯x, ¯y)) ≥d((x, y),(¯x, ¯y)) for (x, y)∼(0,0) and (¯x, ¯y)∼(1,0). Set ψ(t) := φ((0,0),(1, t)). Then ψ∈C2(R), ψ(0) = 1 and ψ(t)≥1 + |t|, which is impossible. 3If f: [0,1] →Ris a 1-Lipschitz map with f(0) = 0 and f(1) = 0, then γ(t) := (t, f(t)) is a length-minimizer from (0,0) to (0,1) for the `∞-distance on R2. Moreover, convergence in W1,∞([0,1]) and in W1,∞([0,1]; R2) are equivalent for such curves. Hence, the set of all lengthminimizers from (0,0) to (0,1) contains as a closed subset the unit ball of W1,∞([0,1]), which is not compact. REGULARITY OF SPHERES 15 Since Sis transversal to the dilations, φis a diffeomorphism. Moreover, if Γ := {(p, d0(p)) : p∈S} ⊂ S×(0,+∞) is the graph of the function d0restricted to S, then Sd=φ(Γ). Thanks to the last remark, the estimate (1.2) follows from the next lemma. Lemma 4.2. Let Ω⊂Rnbe an open set and let f: Ω →Rbe an α-H¨older function, i.e., for all x, y ∈Ωwe have |f(x)−f(y)| ≤ C|x−y|α, for some C > 0, where α∈(0,1]. Define the graph of fas Γf:= {(x, f(x)) : x∈Ω} ⊂ Rn+1. Then n≤dimHΓf≤n+ 1 −α, where dimHis the Hausdorff dimension. Moreover, this estimate is sharp, i.e., there exists fsuch that dimHΓf=n+ 1 −α. The proof is straightforward by use of a simple covering argument or by an estimate of the Minkowski content of the graph. The sharpness of this result has been shown in [5] for the case n= 1. The general case, as stated here, is a simple consequence. Indeed, if g: (0,1) →Ris a α-H¨older function such that dimH(Γg) = 2−α, then the graph of the function f(x1, . . . , xn) := g(x1) is Γf= Γg×(0,1)n−1. Therefore, dimH(Γf) = n+ 1 −α. In the next easy-to-prove lemma we point out that a homogenous distance is locally Lipschitz if and only if the spheres are Lipschitz graphs in the directions of the dilations. Lemma 4.3. Let dbe a homogeneous distance on G. Let Sand Sdbe as in Remark 4.1 and p∈S. Then the following conditions are equivalent: (i) Setting ˆp:= δd0(p)−1(p)∈Sd, the sphere Sdis a Lipschitz graph in the direction ¯ δ(ˆp)in some neighborhood of p; (ii) d0|S:S→(0,+∞)is Lipschitz in some neighborhood of pin S; (iii) d0is Lipschitz in some neighborhood of δλpfor one, hence all, λ > 0. Thanks to Lemma 4.3, Theorem 1.2 is a consequence of Theorem 1.1. 4.2. An intrinsic approach. In this section we will prove Theorem 1.3. We define acone in Rnas Cone(α, h, v) := {x∈Rn:|x| ≤ hand ∠(x, v)≤α} ⊂ Rn, where α∈[0, π], h∈(0,+∞], v∈Rnis the axis of the cone, and ∠(x, v) is the angle between xand v. The following lemma is a simple calculus exercise and it will be used later in the proof of Theorem 1.3. Roughly speaking, it states that a small smooth deformation of a cone still contains a cone with the same tip. Lemma 4.4. Let m, k, n ∈N,p∈Rmand y0∈Rk. Let φ:Rm×Rk→Rnbe a smooth map such that d(φp)(y0) : Rk→Rnis surjective, where φx(y) := φ(x, y). Let C0⊂Rkbe a cone with axis v0∈Rk. Then there exist a cone C⊂Rnwith axis d(φp)(y0)v0and an open neighborhood U⊂Rmof psuch that for all q∈U φq(y0) + C⊂φq(y0+C0). 16 LE DONNE AND NICOLUSSI GOLO Proof of Theorem 1.3. In this proof, we consider the dilations δλas defined for λ≤0 too, with the same definition as for λ > 0. Notice that in this way the map G×R→G, (p, λ)7→ δλp, is a smooth map. Let v1, . . . , vrbe a basis for V1and set pi:= exp(vi)∈G. Up to a rescaling, we can assume d0(pi)<1 for all i. For p∈Gdefine φp:R2r+1 →Gas φp(u1, . . . , ur, s, v1, . . . , vr) = δu1p1···δurpr·δsp·δv1p1···δvrpr. Let ˆx∈R2r+1 be the point with ui= 0, s= 1 and vi= 0, so that φp(ˆx) = p. The differential of φpat ˆxis given by the partial derivatives ∂φp ∂ui (ˆx) = d dt|t=0 (δtpi·p) = dRpd dt|t=0(δtpi)= dRp(vi), ∂φp ∂s (ˆx) = d dt|t=1 (δtp) = ¯ δ(p), ∂φp ∂vi (ˆx) = d dt|t=0 (p·δtpi) = dLpd dt|t=0(δtpi)= dLp(vi). Therefore, if p∈Sdis such that the condition (1.1) is true, then the differential dφphas full rank at ˆx, hence in a neighborhood of ˆx. Define ∆ := {(u1, . . . , ur, s, v1, . . . , vr)∈R2r+1 :s+ r X i=1 (|ui|+|vi|)≤1}. We identify Gwith Rnthrough an arbitrary diffeomorphism. So, by Lemma 4.4, there is a cone Cwith axis ¯ δ(p) and a neighborhood Uof psuch that for all q∈U q+C⊂φq(∆). Up to restricting U, for all q∈Uthere are cones Cqwith axis ¯ δ(q), fixed amplitude and fixed height such that q+Cq⊂q+C. Notice that for all q∈Sdwe have φq(∆) ⊂¯ Bd(0,1) and φq(∆) ∩Sd={q}. In particular, for all q∈Sd∩U, we have q+Cq⊂¯ Bd(0,1) and (q+Cq)∩Sd={q}, i.e., Sd∩Uis a Lipschitz graph in the direction of the dilations. Thanks to Lemma 4.3, we get that d0is Lipschitz in a neighborhood of p. Finally, some considerations on condition (1.1) are due. Remark 4.5.If (1.1) holds at p∈Gand u∈L∞([0,1]; V1) is a control such that End0(u) = p, then the differential dEnd0(u) is surjective, i.e., pis a regular value of End. Indeed, by [20] (see (2.6) and (2.11) there), we have dLp(V1) + dRp(V1) + span{¯ δ(p)} ⊂ =( dEnd0(u)), because q7→ ¯ δ(q) is a contact vector field of G. Proposition 4.6. Let X∈V1. If (1.1) holds for p= exp(X), then g=V1+ [X, V1]. Proof. Let X1, . . . , Xrbe a basis for V1and Y1, . . . , Y`a basis for [X, V1]. Let αi j∈Rbe such that [X, Xi] = P` j=1 αi jYj. First, notice that T0G= dLexp(−X)dLexp(X)(V1) + dRexp(X)(V1) =V1+ dLexp(−X)◦dRexp(X)(V1) =V1+ Adexp(X)(V1). REGULARITY OF SPHERES 17 Then, using the formula Adexp(X)(Y) = eadX(Y) = P∞ k=0 1 k!adk X(Y), we have Adexp(X)(Xi) = Xi+ ∞ X k=1 1 k!adk−1 X([X, Xi])! =Xi+  ∞ X k=1 adk−1 X( ` X j=1 αi jYj)  =Xi+ ` X j=1 αi j ∞ X k=1 adk−1 X(Yj)!. It follows that dim V1+ Adexp(X)V1≤r+`and therefore dim g≤r+`, i.e., g=V1+ [X, V1].  Proposition 4.7. Let Z∈Vk, where k > 0is such that Vi={0}for all i>k. If (1.1) holds for p= exp(Z), then g=V1+ span{Z}. Proof. Since [Z, g] = {0}, we have Rp=Lp. Moreover, ¯ δ(p) = dLp(kZ). So, condition (1.1) becomes dLp(V1) + dLp(span{Z}) = TpG. In particular, if (1.1) holds for all p∈G\{0}, then g=V1⊕V2with dim V2≤1 and [X, V1] = V2for all non-zero X∈V1. 5. Examples 5.1. Three gradings on R2.We will present three examples of dilations on R2. In particular we want to illustrate two applications of Theorem 1.3 and show the sharpness of the dimension estimate (1.2). In Remark 5.5 we give an easy example of a homogeneous distance whose unit ball is a Lipschitz domain, but the distance is not locally Lipschitz away from the diagonal. The first and the easiest is δλ(x, y) := (λx, λy), which gives rise to the known structure of vector space. Here, homogeneous distances are given by norms and balls are convex, hence Lipschitz domains. It’s trivial to see that condition (1.1) holds for all p∈R2. The second example is given by the dilations δλ(x, y) := (λx, λ2y). In this case, R2=V1⊕V2with V1=R×{0}and V2={0}×R, and ¯ δ(x, y)=(x, 2y). Condition (1.1) holds for all (x, y)∈R2with y6= 0. One can actually show that, for any homogeneous metric on (R2, δλ) with closed unit ball Bcentered at 0, the set I={x∈R: (x, 0) ∈B}is a closed interval and there exists a function f:I→R that is locally Lipschitz on the interior of Isuch that Sd∩{(x, y) : y≥0}={(x, f(x)) : x∈I}. We will prove a similar statement in the Heisenberg group with an argument that applies here too, see Section 6. The third example is given by the dilations (5.1) δλ(x, y) := (λ2x, λ2y), 18 LE DONNE AND NICOLUSSI GOLO and it is interesting because of the next proposition. Proposition 5.1. There exists a homogeneous (with respect to dilations (5.1)) distance don R2whose unit sphere has Euclidean Hausdorff dimension 3 2. Notice that 3 2is the maximal Hausdorff dimension that one gets by the estimate (1.2). For proving Proposition 5.1, we need to find a set B⊂R2that satisfies all four conditions listed in Lemma 2.15, in particular (5.2) ∀p, q ∈B, ∀t∈[0,1] t2p+ (1 −t)2q∈B. One easily proves the following preliminary facts. Lemma 5.2. Let p, q ∈R2and γ: [0,1] →R2,γ(t) := t2p+ (1 −t)2q. (1) The curve γis contained in the triangle of vertices 0, p, q. (2) The curve γis an arc of the parabola passing through pand qand that is tangent to the lines span{p}and span{q}. (3) If Bsatisfies (5.2) and A:R2→R2is a linear map, then A(B)satisfies (5.2) as well. Lemma 5.3. For 0< C ≤1, define YC:= {(x, y) : |x| ≤ 1, y ≤1 + Cp|x|}. Then YCsatisfies (5.2). Proof. Let p, q ∈YCand set γ(t) = (γx(t), γy(t)) := t2p+ (1 −t)2q. If both pand qstay on one side with respect to the vertical axis, then γ(t)∈YC for all t∈[0,1] thanks to the first point of Lemma 5.2 and because the two sets YC∩{x≥0}and YC∩{x≤0}are convex. Therefore, we suppose that p= (−px, py)q= (qx, qy) with px, qx>0. Let t0∈[0,1] be the unique value such that γx(t0) = 0. Then the curve γlies in the union of the two triangles with vertices 0, γ(0), γ(t0) and 0, γ(1), γ(t0), respectively. Therefore, γlies in YCif and only if γy(t0)≤1. Solving the equation γx(t0) = t2 0(qx−px)−2qxt0+qx= 0, one gets t0=√qx √qx+√px ,(1 −t0) = √px √qx+√px . From the expression of γy(t0) = t2 0py+ (1 −t0)2qy, we notice that, pxand qxfixed, the worst case is when pyand qyare maximal, i.e., py= 1 + C√px, qy= 1 + C√qx. REGULARITY OF SPHERES 19 Finally γy(t0) = t2 0py+ (1 −t0)2qy =qx (√qx+√px)2(1 + C√px) + px (√qx+√px)2(1 + C√qx) =1 (√qx+√px)2(qx+px+Cqx√px+Cpx√qx) = 1 + −2√pxqx+Cqx√px+Cpx√qx (√qx+√px)2 = 1 + √pxqx−2 + C(√qx+√px) (√qx+√px)2. Since −2 + C(√qx+√px)≤0, then we have γy(t0)≤1, as desired.  Lemma 5.4. Let α, β > 0. For all 0<  ≤αand all 0< C ≤β √α, the set Y(, β, C) := {(x, y) : |x| ≤ , y ≤β+Cp|x| satisfies (5.2). Proof. Define the linear map A(x, y) := (αx, βy) and set C0:= C√α β≤1. Then one just needs to check that Y(, β, C) = A(YC0)∩{(x, y) : |x| ≤ }, where YC0is defined as in the previous Lemma 5.3.  Proof of Proposition 5.1. First of all, let θ0>0 be such that for all |θ| ≤ θ0it holds (5.3) |θ| 2≤ |cos(π 2+θ)|=|sin θ| ≤ 2|θ|. Moreover, let L, m, M, C > 0 be such that L√2 √m≤C≤m √2Mθ0 . Let f:R→(0,+∞) be a function such that (5.4) ∀s, t ∈R|f(t)−f(s)| ≤ Lp|t−s|, (5.5) ∀t∈Rm≤f(t)≤M. We claim that, for |θ| ≤ θ0, we have (5.6) fπ 2+θ·cos(π 2+θ),sin(π 2+θ)∈Y2Mθ0, f(π 2), C where Y2Mθ0, f(π 2), Cis defined as in Lemma 5.4. Indeed, we have on one side |x|:= |f(π 2+θ) cos(π 2+θ)| ≤ M2|θ| ≤ 2Mθ0. 20 LE DONNE AND NICOLUSSI GOLO On the other side, y:= f(π 2+θ) sin(π 2+θ)≤f(π 2+θ) ≤f(π 2) + f(π 2+θ)−f(π 2)≤f(π 2) + Lp|θ| ≤f(π 2) + Lp2 cos(π 2+θ)f(π 2+θ) pf(π 2+θ)≤f(π 2) + √2L √mp|x| ≤f(π 2) + Cp|x|. So (5.6) is satisfied. Since for α:= 2Mθ0and β:= f(π 2) we have β √α=f(π 2) √2Mθ0≥m √2Mθ0≥C, Lemma 5.4 applies and we get that Y2Mθ0, f(π 2), Csatisfies (5.2). For any θwe set Aθto be the counterclockwise rotation of angle θ: Aθ=cos θ−sin θ sin θcos θ. Define the curve φ(t) := f(t)(cos t, sin t). Notice that Aθφ(t) = f((t−θ)+θ)(cos(t+ θ),sin(t+θ)) and that the function s7→ f(s+θ) is still satisfying both (5.4) and (5.5). So we have that, for |t|,|s|<θ0 2, φ(π 2+t)∈As[Y(2Mθ0, f(π 2+s), C)] and the set As[Y(2Mθ0, f(π 2+s), C)] satisfies (5.2). Set B:= \ |s|<θ0 2As[Y(2Mθ0, f(π 2+s), C)] ∩−As[Y(2Mθ0, f(π 2+s), C)]. The set Bsatisfies all three conditions of Lemma 2.15, hence it is the unit ball of a homogeneous metric. Moreover, {φ(π 2+t) : |t|<θ0 2} ⊂ ∂B. The statement of Proposition 5.1 follows because there are functions f:R→ [0,+∞) that satisfy (5.4) and (5.5) and such that the image of the curve φhas Hausdorff dimension 3 2. Indeed, the image of φhas the same Hausdorff dimension of the graph of f, and then one uses the sharpness of Lemma 4.2.  Remark 5.5.Using the same arguments as in the proof of Lemma 5.3, one easily shows that the set B:= {(x, y)∈R2:|x| ≤ 1,−f(−x)≤y≤f(x)}, where f(x) := (1x≤0 1 + √x x > 0, is the unit ball of a homogeneous distance on R2with dilations (5.1). Notice that such Bis a Lipschitz domain, but the associated homogeneous distance is not Lipschitz in any neighborhood of the point (0,1), thanks to Lemma 4.3. REGULARITY OF SPHERES 21 5.2. The Heisenberg groups. In the Heisenberg groups Hn(for an introduction see [10]) condition (1.1) holds at every non-zero point. Therefore, balls of any homogeneous metric on Hnare Lipschitz domains. We will treat more in detail the first Heisenberg group in Section 6. 5.3. A sub-Finsler sphere with a cusp. Let Hbe the first Heisenberg group (see Section 6 for the definition). The group G=H×Ris a stratified group with grading (V1×R)⊕V2, where V1⊕V2is a stratification for H. The line {0H}×R is a singular curve in G. Moreover, it has been shown in [8] that there exists a sub-Finsler distance on Gwhose unit sphere Sdhas a cusp in the intersection Sd∩({0H}×R). However, for sub-Riemannian metrics we still have balls that are Lipschitz domains, as the following Proposition 5.7 shows. But let us first recall a simple fact: Lemma 5.6. Let Aand Bbe two stratified groups with stratifications LViand LWi, respectively. Endow V1and W1with a scalar product each and let dA, dBbe the corresponding homogeneous sub-Riemannian distances. Then A×Bis a Carnot group with stratification LiVi×Wiand metric d((a, b),(a0, b0)) := pdA(a, a0)2+dB(b, b0)2, which is the sub-Riemannian metric generated by the scalar product on V1×W1 induced by the scalar products on V1and W1. One proves this lemma by using the fact that the energy of curves on A×B(i.e., the integral of the squared norm of the derivative) is the sum of the energies of the two components of the curve. Proposition 5.7. Any homogeneous sub-Riemannian metric on H×Ris locally Lipschitz away from the diagonal. Proof. First of all, we show that, up to isometry, there is only one homogeneous sub-Riemannian distance on H×R. Let (X1, Y1, T1) and (X2, Y2, T2) be two bases of V1×Rthat are orthonormal for two sub-Riemannian structures, respectively. We may assume T1, T2∈ {0}×R. Notice that [Xi, Yi]/∈V1×R. The linear map such that X17→ X2,Y17→ Y2,T17→ T2, [X1, Y1]7→ [X2, Y2] is an automorphism of Lie algebras and induces an isometry between the two sub-Riemannian structures. Denoting by dHand dRthe standard metrics on Hand R, respectively, we prove the proposition for the product metric as in Lemma 5.6. Namely, we need to check that the function (5.7) (p, t)7→ d((0,0),(p, t)) = pdH(0, p)2+t2 is locally Lipschitz at all (ˆp, ˆ t)6= (0,0). This follows directly from Proposition 3.3.  5.4. A sub-Riemannian sphere with a cusp. Proposition 5.8. Let Gbe a Carnot group of step 3 endowed with a sub-Riemannian distance dG. Then the sub-Riemannian distance don G×Rgiven by d((p, s),(q, t)) = pdE(p, q)2+|t−s|2 has a unit sphere with a cusp at (0G,1). 22 LE DONNE AND NICOLUSSI GOLO Proof. Let g=L3 i=1 Vibe the Lie algebra of Gand fix Z∈V3\{0}. We identify gwith Gthrough the exponential map. The intersection of the unit sphere in (G×R, d) with the plane span{Z}×Ris given by all points (zZ, t) such that (5.8) dG(0, zZ)2+t2= 1. Since dGis homogeneous on G, there exists C > 0 such that for all z∈R (5.9) dG(0, zZ) = C|z|1 3. Putting together (5.8) and (5.9) we obtain that this intersection consists of all the points (zZ, t) such that |z|=1 + t C23 2 ·(1 −t)3 2. One then easily sees that this set in R2has a cusp at (0,1).  6. A closer look at the Heisenberg group The Heisenberg group His the easiest example of a stratified group that is not Abelian and for this reason it has been studied in large extend. The most common homogeneous metrics on Hare the Kor´anyi metric and the sub-Riemannian metric. Sub-Finsler metrics on Harise in the study of finitely-generated groups, see [7] and references therein. The geometry of sub-Finsler spheres has been studied in [21] and [14]. The Lie algebra hof the Heisenberg group is a three dimensional vector space span{X, Y, Z}with a Lie bracket operation defined by the only nontrivial relation [X, Y ] = Z. We identify the Heisenberg group Hagain with span{X, Y, Z}, where we define the group operation p·q:= p+q+1 2[p, q]∀p, q ∈H. Hence his the Lie algebra of Hand the exponential map h→His the identity map. Notice that the inverse of an element pis p−1=−p. The Heisenberg Lie algebra admits the stratification h=V1⊕V2with V1= span{X, Y }and V2= span{Z}. Denote by πthe linear projection h→V1along V2. Notice that this map, regarded as π: (H,·)→(V1,+), is a group morphism. The dilations δλ:H→Hare explicitly expressed by δλ(xX +yY +zZ) = xλX +yλY +zλ2Z, ∀λ > 0. These are both Lie algebra automorphisms δλ:h→hand Lie group automorphisms H→H. Three are the main results of this section. Proposition 6.1. Let N:H→[0,+∞)be a homogeneous norm. Then the unit ball B:= {p∈H:N(p)≤1} is a star-like Lipschitz domain. REGULARITY OF SPHERES 23 Proof. One easily shows that condition (1.1) holds for all p∈H\ {0}. In order to prove that Bis star-like, one first notice that if p∈B, then −p∈B, hence δt(p)δ1−t(−p) = (2t−1)p∈Bfor all t∈[0,1], and this is a straight line passing through zero.  Proposition 6.2. Let Nand Bas in Proposition 6.1. Set K:= π(B)⊂V1. Then Kis a compact, convex set with K=−Kand K= cl(int(K)), and there exists a function f:K→[0,+∞), locally Lipschitz on int(K), such that (6.1) B={v+zZ :v∈K, −f(−v)≤z≤f(v)}. The proof is postponed to Section 6.1. We remark that homogeneous distances and sub-Finsler homogeneous distances on Hhave a precise relation. Indeed, if dis a homogeneous distance on H, then it is easy to show that the length distance generated by dis exactly the sub-Finsler distance that has the norm on V1generated by the set Kdefined in Proposition 6.2. Proposition 6.3. Let K⊂V1be a compact, convex set with −K=Kand 0∈ int(K). Let g:K→Rbe Lipschitz. Then there exists b∈Rsuch that for f:= g+b the set Bas in (6.1) is the unit ball of a homogeneous norm. The proof will appear in Section 6.2. As a consequence of Proposition 6.3, we get the existence of homogeneous distances on Hthat are not almost convex in the sense of [13]. Indeed, one can take the distance associated to g(xX +yY ) = |x|from Proposition 6.3. 6.1. Proof of Proposition 6.2. Lemma 6.4. Let B⊂Hbe an arbitrary closed set satisfying (2.5). If p=v+zZ ∈ Bwith v=π(p)∈V1, then v+szZ ∈B, for all s∈[0,1]. In particular, (1) π(B) = B∩V1; (2) π(B)⊂V1is convex. Proof. We have that for all t∈[0,1] B3δtp·δ1−tp=v+ (t2+ (1 −t)2)zZ. Since the image of [0,1] through the map t7→ (t2+(1 −t)2) is [1 2,1], then it follows v+szZ ∈Bfor all s∈[1 2,1]. Iterating this process and using the closeness of B, we get v+szZ ∈Bfor all s∈[0,1]. For the last statement, take v, w ∈π(B)⊂B and notice that tv + (1 −t)w=π(δtv·δ1−tw)∈π(B).  Let B={N≤1}be the unit ball of a homogeneous norm and set K:= π(B)⊂ V1and Ω := int(K). First, we check that ¯ Ω = K. On the one hand, clearly we have ¯ Ω⊂K. On the other hand, if v∈K, then for any t∈[0,1) we have N(δtv) = tN(v)<1, i.e., δtv=tv ∈intB∩V⊂Ω. Hence v∈¯ Ω. If we define f:K→[0,+∞) as f(v) := max{z:v+zZ ∈Q}, then we have (6.1). In order to prove that fis locally Lipschitz on Ω, we need to prove (6.2) ∀p∈∂B ∩{z≥0}∩π−1(Ω), ∃U3popen ,∃Cvertical cone, s.t. ∀q∈U∩∂B it holds q+C⊂B. Here a vertical cone is a Euclidean cone with axis −Zand non-empty interior. 24 LE DONNE AND NICOLUSSI GOLO So, fix p∈∂Q ∩{z≥0}such that π(p)∈Ω. Define for θ∈Rand  > 0 vθ:= xθX+yθY:= cos(θ)X+sin(θ)Y. For  > 0 small enough, π(p) + vθ∈Ω for all θ. Define φ(t, θ) := δ(1−t)p·δt(π(p) + vθ). Clearly φ(t, θ)∈Bfor t∈[0,1] and θ∈R, and φ(0, θ) = pfor all θ. Geometrically, φ([0,1] ×R) is a “tent” inside Bstanding above the whole vertical segment from π(p) to p. Notice that p6=π(p), indeed N(p) = 1 while N(π(p)) <1, because π(p)∈Ω. We only need to prove that the curves t7→ φ(t, θ) meet this vertical segment by an angle bounded away from 0. Some computations are needed: set p=p1X+ p2Y+p3Z, then φ(t, θ) = π(p) + tvθ+1 2t(1 −t)(p1yθ−p2xθ) + (1 −t)2p3Z. We take care only of the third coordinate. Set g(t):=1 2t(1 −t)(p1yθ−p2xθ) + (1 −t)2p3 =t2(−1 2(p1yθ−p2xθ) + p3) + t(1 2(p1yθ−p2xθ)−2p3) + p3. Saying that the angle between the curve t7→ φ(t, θ) and the vertical segment at pis uniformly grater than zero, is equivalent to give an upper bound to the derivative of gat 0 for all θ. Since g0(0) = 1 2(p1yθ−p2xθ)−2p3, we are done. Finally, since both and g0(0) depend continuously on p, then (6.2) is satisfied.  6.2. Proof of Proposition 6.3. We consider the bilinear map ω:V1×V1→R given by ω(v1X+v2Y, w1X+w2Y) := v1w2−v2w1. Lemma 6.5. For any continuous function f:K→[0,+∞), the set Bas in (6.1) is the unit ball of a homogeneous norm on Hif and only if (6.3) ∀v, w ∈K∀t∈[0,1] f(tv + (1 −t)w)−t2f(v)−(1 −t)2f(w)−t(1−t) 2ω(v, w)≥0. Proof. One easily sees that B=B−1. Notice that Bis the unit ball of a homogeneous norm if and only if it satisfies (2.5). ⇒Assume that Bsatisfies (2.5). Then for any v, w ∈Kwe have B3δt(v+f(v)Z)·δ(1−t)(w+f(w)Z) = =tv + (1 −t)w+t2f(v) + (1 −t)2f(w) + 1 2t(1 −t)ω(v, w)Z, hence t2f(v) + (1 −t)2f(w) + 1 2t(1 −t)ω(v, w)≤f(tv + (1 −t)w).