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1 SOME TOPICS OF GEOMETRIC MEASURE THEORY IN CARNOT GROUPS DRAFT FRANCESCO SERRA CASSANO Contents 1. Introduction 1 2. An introduction to Carnot Groups 2 2.1. Definition and first properties. 2 2.2. Metrics on Carnot groups 5 2.3. Hausdor↵measures in a metric space. Applications to Carnot groups. 9 2.4. Carnot groups as tangent spaces to sub-Riemannian manifolds. 17 3. Di↵erential Calculus on Carnot Groups 17 3.1. Pansu di↵erentiable maps between Carnot groups and some their characterizations 17 3.2. Area formula between Carnot groups 21 3.3. Whitney theorem for maps between Carnot groups 21 3.4. Sobolev spaces and Poincar´einequality. 22 3.5. Vector fields. 22 3.6. Sobolev spaces associated with vector fields. 23 3.7. Poincar´einequality. 23 3.8. BV -functions and sets of finite perimeter on Carnot groups. 26 4. Di↵erential Calculus within Carnot Groups 28 4.1. Complementary subgroups and graphs 29 4.2. Intrinsic graphs in Carnot groups 34 4.3. Regular hypersurfaces in Carnot groups. 36 4.4. Area formulas for Hausdor↵measures in metric spaces. Application to Carnot groups. 40 4.5. Regular surfaces in Carnot groups 44 4.6. Intrinsic Lipschitz Graphs in Carnot groups 48 4.7. Intrinsic di↵erentiable graphs in Carnot groups 53 4.8. H-regular surfaces, intrinsic Lipschitz graphs in Heisenberg groups vs. intrinsic di↵erentiability 57 4.9. Rectifiability in Carnot groups. 60 5. Sets of finite perimeter and minimal surfaces 64 F.S.C. is supported by MURST and INDAM, Italy and University of Trento, Italy. 1 5.1. De Giorgi’s structure theorem for sets of finite perimeter in Carnot groups of step 2 64 5.2. Minimal boundaries in Carnot groups 70 References 70 1. Introduction These notes aim at illustrating some results achieved in the geometric measure theory in Carnot groups. First of all, I would like to thank the organizers of the ”Geome- try, Analysis and Dynamics on sub-Riemannian Manifolds” trimester, which is being held in Paris, for their kind invitation. It is also a great pleasure for me to acknowledge the help and support of several friends of mine who made this work possible: first of all, most of the results I obtained and that I presented here have been achieved jointly with Bruno Franchi and Raul Serapioni. Our long collaboration has been always an invaluable source of scientific as well as human enrichment. Without their collaboration and friendship, I would never have been able to attack this hard subject. I have to thank them also for allowing me to widely quote both our joint papers and their latest ones. I have also to thank the support of other friends and collaborators with whom I shared fruitful discussions during the last 15 years and whose work is mentioned here : A. Agrachev, G. Alberti, L. Ambrosio, Z. Balogh, S. Bianchini, F. Bigolin,U. Boscain, L. Capogna, L. Car- avenna, J.H. Cheng, G. Citti, B. Kirchheim, E. Le Donne, N. Garo- falo, P. Haj lasz, P. Koskela, B. Kleiner,G.P. Leonardi, V. Magnani, A. Malchiodi, P. Mattila, F. Montefalcone, M. Miranda jr., R. Monti, P. Pansu, S. Pauls, A. Pinamonti, S. Rigot, M. Ritor´e, J. Tyson, D. Vit- tone, P. Yang........and many others whom I am likely to have left out, for which I apologize. These notes are not intended to provide a thorough – not even a partial – survey of the geometric measure theory in Carnot groups, since they are meant to be presented in a few lectures during this Paris trimester. Anyone interested in an exhaustive overview of this subject with a full bibliography, sharp statements, and detailed proofs, may refer to the PhD theses of V. Magnani [143], R. Monti [170] and F. Montefal- cone [165], as well as to the recent ones of F. Bigolin [35], D. Vittone [206], A. Pinamonti [185]. For more specific aspects we restrict our- selves to recommend the reader to the general monographes [71], [120], [121], [117], [116], [202], [205], [169], [49], [42], [2] and to the papers [7], [8], [10], [47], [66], [96], [102], [103][111], [122] [132], [180], [179], [181], [182], [208] and to the references therein. 2 Since these lectures especially focus on some issues of geometric mea- sure theory in Carnot groups, they of course leave out many other important topics, such as, for instance, coarea formulas (see[143]), cur- rents and Rumin’s complex (see f [8],[103],[193],[194],[18] and [107]), curvatures (see [49],[2],[127],[188] and [31]), mass transportation and optimal transport (see [11], [190],[81] and[76]). 2. An introduction to Carnot Groups 2.1. Definition and first properties. The present subsection is largely taken from [102] and [99] (see also [101]). A Carnot group G of step k (see [82], [126], [174], [121], [179], [204], [205], [42]) is a connected, simply connected Lie group whose Lie algebra g admits a step k strat- ification, i.e. there exist linear subspaces V1, ..., Vk such that (1) g = V ... V , [V ,V]=V ,V= 0 ,V= 0 if i>k, 1 ⊕ ⊕ k 1 i i+1 k 6 { } i { } where [V1,Vi] is the subspace of g generated by the commutators [X, Y ] with X V1 and Y Vi.Letmi = dim Vi,fori =1,...,k and h = m 2+ + m with2 h = 0 and, clearly, h = n. Choose a basis i 1 ··· i 0 k e1,...,en of g adapted to the stratification, that is such that ehj 1+1,...,ehj is a base of Vj for each j =1,...,k. − Let X = X1,...,Xn be the family of left invariant vector fields such that Xi(0) = ei. Given (1), the subset X1,...,Xm1 generates by com- mutations all the other vector fields; we will refer to X1,...,Xm1 as generating vector fields of the group. The exponential map is a one to one map from g onto G, i.e. any p G can be written in a unique 2 way as p = exp(p1X1 + + pnXn). Using these exponential coordi- ··· n nates, we identify p with the n-tuple (p1,...,pn) R and we identify 2 G with (Rn, ) where the explicit expression of the group operation is determined· by the Campbell-Hausdor↵formula (see [42]) and some· of its features are described in the following Proposition 2.3. If p G i mi 2 and i =1,...,k, we put p =(phi 1+1,...,phi ) R , so that we can − 2 also identify p with [p1,...,pk] Rm1 Rmk = Rn. 2 ⇥···⇥ The subbundle of the tangent bundle T G that is spanned by the vector fields X1,...,Xm1 plays a particularly important role in the theory, it is called the horizontal bundle HG; the fibers of HG are HGx = span X1(x),...,Xm1 (x) ,xG. { } 2 A subriemannian structure is defined on G, endowing each fiber of HG with a scalar product , x and with a norm x that make h· ·i |·| the basis X1(x),...,Xm1 (x) an orthonormal basis. That is if v = m1 m1 i=1 viXi(x)=(v1,...,vm1 ) and w = i=1 wiXi(x)=(w1,...,wm1 ) m1 2 are in HGx, then v, w x := vjwj and v x := v, v x. P h i j=1 P | | h i The sections of HG are called horizontal sections,avectorofHGx is P an horizontal vector while any vector in T Gx that is not horizontal is 3 a vertical vector. Each horizontal section is identified by its canoni- cal coordinates with respect to this moving frame X1(x),...,Xm1 (x). This way, an horizontal section ' is identified with a function ' = n m1 ('1,...,'m1 ):R R . When dealing with two such sections ' and whose argument! is not explicitly written, we drop the index x in the scalar product writing ,' for (x),'(x) x. The same convention is adopted for the norm.h i h i Two important families of automorphism of G are the so called in- trinsic translations and the intrinsic dilations of G.Foranyx G, the 2 (left) translation ⌧x : G G is defined as ! z ⌧ z := x z. 7! x · For any λ>0, the dilation δλ : G G, is defined as ! ↵1 ↵n (2) δλ(x1, ..., xn)=(λ x1, ..., λ xn), where ↵i N is called homogeneity of the variable xi in G (see [83] Chapter 1)2 and is defined as (3) ↵j = i whenever hi 1 +1 j hi, − hence 1 = ↵ = ... = ↵ <↵ =2 ... ↵ = k. 1 m1 m1+1 n Example 2.1 (Euclidean spaces). The Euclidean space Rn can be meant a (trivial) abelian step 1 Carnot group with respect to its struc- ture of vector space. Namely the group law can be defined by the standard sum p + q, if p, q Rn, and the dilations defined λp, if 2 p Rn and λ>0. The standard base of the Lie algebra E of Rn 2 is given by the left-invariant vector fields Xj = @j, j =1,...,n. All commutator relations in E are trivial. Example 2.2 (Heisenberg group). The simplest example of Carnot group is provided by the Heisenberg group Hn = R2n+1. The reader can find an exhaustive introduction to this structure as well as applications n 2n+1 in [202].A point p H is denoted by p =(p1,...,p2n,p2n+1) R . 2 2 For p, q Hn, we define the group operation given by the standard sum p + q2 1 2n p q = p + q ,...,p + q ,p + q + (p q p q ) . · 1 1 2n 2n 2n+1 2n+1 2 i i+n − i+n i i=1 ! X and the family of (non isotropic) dilations 2 n δλ(p) := (λp1,...,λp2n,λ p2n+1) p H ,λ>0 .
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