DOCSLIB.ORG

1 Some Topics of Geometric Measure Theory in Carnot Groups Draft

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
1 Some Topics of Geometric Measure Theory in Carnot Groups Draft 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 .
Load more

Recommended publications
  • Rigidity of Quasiconformal Maps on Carnot Groups
    RIGIDITY OF QUASICONFORMAL MAPS ON CARNOT GROUPS Mark Medwid A Dissertation Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY August 2017 Committee: Xiangdong Xie, Advisor Alexander Tarnovsky, Graduate Faculty Representative Mihai Staic Juan Bes´ Copyright c August 2017 Mark Medwid All rights reserved iii ABSTRACT Xiangdong Xie, Advisor Quasiconformal mappings were first utilized by Grotzsch¨ in the 1920’s and then later named by Ahlfors in the 1930’s. The conformal mappings one studies in complex analysis are locally angle-preserving: they map infinitesimal balls to infinitesimal balls. Quasiconformal mappings, on the other hand, map infinitesimal balls to infinitesimal ellipsoids of a uniformly bounded ec- centricity. The theory of quasiconformal mappings is well-developed and studied. For example, quasiconformal mappings on Euclidean space are almost-everywhere differentiable. A result due to Pansu in 1989 illustrated that quasiconformal mappings on Carnot groups are almost-everywhere (Pansu) differentiable, as well. It is easy to show that a biLipschitz map is quasiconformal but the converse does not hold, in general. There are many instances, however, where globally defined quasiconformal mappings on Carnot groups are biLipschitz. In this paper we show that, under cer- tain conditions, a quasiconformal mapping defined on an open subset of a Carnot group is locally biLipschitz. This result is motivated by rigidity results in geometry (for example, the theorem by Mostow in 1968). Along the way we develop background material on geometric group theory and show its connection to quasiconformal mappings. iv ACKNOWLEDGMENTS I would like to acknowledge, first and foremost, my wife, Heather.
    [Show full text]
  • Casimir Functions of Free Nilpotent Lie Groups of Steps Three and Four
    Casimir functions of free nilpotent Lie groups of steps three and four∗ A. V. Podobryaev A.K. Ailamazyan Program Systems Institute of RAS [email protected] June 2, 2020 Abstract Any free nilpotent Lie algebra is determined by its rank and step. We consider free nilpotent Lie algebras of steps 3, 4 and corresponding connected and simply connected Lie groups. We construct Casimir functions of such groups, i.e., invariants of the coadjoint representation. For free 3-step nilpotent Lie groups we get a full description of coadjoint orbits. It turns out that general coadjoint orbits are affine subspaces, and special coadjoint orbits are affine subspaces or direct products of nonsingular quadrics. The knowledge of Casimir functions is useful for investigation of integration properties of dynamical systems and optimal control problems on Carnot groups. In particular, for some wide class of time-optimal problems on 3-step free Carnot groups we conclude that extremal controls corresponding to two-dimensional coad- joint orbits have the same behavior as in time-optimal problems on the Heisenberg group or on the Engel group. Keywords: free Carnot group, coadjoint orbits, Casimir functions, integration, geometric control theory, sub-Riemannian geometry, sub-Finsler geometry. AMS subject classification: 22E25, 17B08, 53C17, 35R03. Introduction The goal of this paper is a description of Casimir functions and coadjoint orbits for arXiv:2006.00224v1 [math.DG] 30 May 2020 some class of nilpotent Lie groups. This knowledge is important for the theory of left- invariant optimal control problems on Lie groups [1]. Casimir functions are integrals of the Hamiltonian system of Pontryagin maximum principle.
    [Show full text]
  • Proving the Regularity of the Reduced Boundary of Perimeter Minimizing Sets with the De Giorgi Lemma
    PROVING THE REGULARITY OF THE REDUCED BOUNDARY OF PERIMETER MINIMIZING SETS WITH THE DE GIORGI LEMMA Presented by Antonio Farah In partial fulfillment of the requirements for graduation with the Dean's Scholars Honors Degree in Mathematics Honors, Department of Mathematics Dr. Stefania Patrizi, Supervising Professor Dr. David Rusin, Honors Advisor, Department of Mathematics THE UNIVERSITY OF TEXAS AT AUSTIN May 2020 Acknowledgements I deeply appreciate the continued guidance and support of Dr. Stefania Patrizi, who supervised the research and preparation of this Honors Thesis. Dr. Patrizi kindly dedicated numerous sessions to this project, motivating and expanding my learning. I also greatly appreciate the support of Dr. Irene Gamba and of Dr. Francesco Maggi on the project, who generously helped with their reading and support. Further, I am grateful to Dr. David Rusin, who also provided valuable help on this project in his role of Honors Advisor in the Department of Mathematics. Moreover, I would like to express my gratitude to the Department of Mathematics' Faculty and Administration, to the College of Natural Sciences' Administration, and to the Dean's Scholars Honors Program of The University of Texas at Austin for my educational experience. 1 Abstract The Plateau problem consists of finding the set that minimizes its perimeter among all sets of a certain volume. Such set is known as a minimal set, or perimeter minimizing set. The problem was considered intractable until the 1960's, when the development of geometric measure theory by researchers such as Fleming, Federer, and De Giorgi provided the necessary tools to find minimal sets.
    [Show full text]
  • Non-Minimality of Corners in Subriemannian Geometry
    NON-MINIMALITY OF CORNERS IN SUBRIEMANNIAN GEOMETRY EERO HAKAVUORI AND ENRICO LE DONNE Abstract. We give a short solution to one of the main open problems in subrie- mannian geometry. Namely, we prove that length minimizers do not have corner- type singularities. With this result we solve Problem II of Agrachev's list, and provide the first general result toward the 30-year-old open problem of regularity of subriemannian geodesics. Contents 1. Introduction 1 1.1. The idea of the argument 2 1.2. Definitions 3 2. Preliminary lemmas 4 3. The main result 6 3.1. Reduction to Carnot groups 6 3.2. The inductive non-minimality argument 7 References 9 1. Introduction One of the major open problems in subriemannian geometry is the regularity of length-minimizing curves (see [Mon02, Section 10.1] and [Mon14b, Section 4]). This problem has been open since the work of Strichartz [Str86, Str89] and Hamenst¨adt [Ham90]. Contrary to Riemannian geometry, where it is well known that all length minimizers are C1-smooth, the problem in the subriemannian case is significantly more difficult. The primary reason for this difficulty is the existence of abnormal curves (see [AS04, Date: March 8, 2016. 2010 Mathematics Subject Classification. 53C17, 49K21, 28A75. Key words and phrases. Corner-type singularities, geodesics, sub-Riemannian geometry, Carnot groups, regularity of length minimizers. 1 2 EERO HAKAVUORI AND ENRICO LE DONNE ABB15]), which we know may be length minimizers since the work of Montgomery [Mon94]. Nowadays, many more abnormal length minimizers are known [BH93, LS94, LS95, GK95, Sus96]. Abnormal curves, when parametrized by arc-length, need only have Lipschitz- regularity (see [LDLMV14, Section 5]), which is why, a priori, no further regular- ity can be assumed from an arbitrary length minimizer in a subriemannian space.
    [Show full text]
  • Master of Science in Advanced Mathematics and Mathematical
    Master of Science in Advanced Mathematics and Mathematical Engineering Title: Delaunay cylinders with constant non-local mean curvature Author: Marc Alvinyà Rubió Advisor: Xavier Cabré Vilagut Department: Department of Mathematics Academic year: 2016-2017 I would like to thank my thesis advisor Xavier Cabr´efor his guidance and support over the past three years. I would also like to thank my friends Tom`asSanz and Oscar´ Rivero for their help and interest in my work. Abstract The aim of this master's thesis is to obtain an alternative proof, using variational techniques, of an existence 2 result for periodic sets in R that minimize a non-local version of the classical perimeter functional adapted to periodic sets. This functional was introduced by D´avila,Del Pino, Dipierro and Valdinoci [20]. Our 2 minimizers are periodic sets of R having constant non-local mean curvature. We begin our thesis with a brief review on the classical theory of minimal surfaces. We then present the non-local (or fractional) perimeter functional. This functional was first introduced by Caffarelli et al. [15] to study interphase problems where the interaction between particles are not only local, but long range interactions are also considered. Additionally, also using variational techniques, we prove the existence of solutions for a semi-linear elliptic equation involving the fractional Laplacian. Keywords Minimal surfaces, non-local perimeter, fractional Laplacian, semi-linear elliptic fractional equation 1 Contents Chapter 1. Introduction 3 1.1. Main result 4 1.2. Outline of the thesis 5 Chapter 2. Classical minimal surfaces 7 2.1. Historical introduction 7 2.2.
    [Show full text]
  • Measure Contraction Properties of Carnot Groups Luca Rizzi
    Measure contraction properties of Carnot groups Luca Rizzi To cite this version: Luca Rizzi. Measure contraction properties of Carnot groups . Calculus of Variations and Partial Differential Equations, Springer Verlag, 2016, 10.1007/s00526-016-1002-y. hal-01218376v3 HAL Id: hal-01218376 https://hal.archives-ouvertes.fr/hal-01218376v3 Submitted on 21 May 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. MEASURE CONTRACTION PROPERTIES OF CARNOT GROUPS RIZZI LUCA Abstract. We prove that any corank 1 Carnot group of dimension k + 1 equipped with a left-invariant measure satisfies the MCP(K, N) if and only if K ≤ 0 and N ≥ k + 3. This generalizes the well known result by Juillet for the Heisenberg group Hk+1 to a larger class of structures, which admit non-trivial abnormal minimizing curves. The number k + 3 coincides with the geodesic dimension of the Carnot group, which we define here for a general metric space. We discuss some of its properties, and its relation with the curvature exponent (the least N such that the MCP(0,N) is satisfied). We prove that, on a metric measure space, the curvature exponent is always larger than the geodesic dimension which, in turn, is larger than the Hausdorff one.
    [Show full text]
  • On Logarithmic Sobolev Inequalities for the Heat Kernel on The
    ON LOGARITHMIC SOBOLEV INEQUALITIES FOR THE HEAT KERNEL ON THE HEISENBERG GROUP MICHEL BONNEFONT, DJALIL CHAFAÏ, AND RONAN HERRY Abstract. In this note, we derive a new logarithmic Sobolev inequality for the heat kernel on the Heisenberg group. The proof is inspired from the historical method of Leonard Gross with the Central Limit Theorem for a random walk. Here the non commutative nature of the increments produces a new gradient which naturally involves a Brownian bridge on the Heisenberg group. This new inequality contains the optimal logarithmic Sobolev inequality for the Gaussian distribution in two dimensions. We compare this new inequality with the sub- elliptic logarithmic Sobolev inequality of Hong-Quan Li and with the more recent inequality of Fabrice Baudoin and Nicola Garofalo obtained using a generalized curvature criterion. Finally, we extend this inequality to the case of homogeneous Carnot groups of rank two. Résumé. Dans cette note, nous obtenons une inégalité de Sobolev logarithmique nouvelle pour le noyau de la chaleur sur le groupe de Heisenberg. La preuve est inspirée de la méthode historique de Leonard Gross à base de théorème limite central pour une marche aléatoire. Ici la nature non commutative des incréments produit un nouveau gradient qui fait intervenir naturellement un pont brownien sur le groupe de Heisenberg. Cette nouvelle inégalité contient l’inégalité de Sobolev logarithmique optimale pour la mesure gaussienne en deux dimensions. Nous comparons cette nouvelle inégalité avec l’inégalité sous-elliptique de Hong-Quan Li et avec les inégalités plus récentes de Fabrice Beaudoin et Nicola Garofalo obtenues avec un critère de courbure généralisé.
    [Show full text]
  • Maximum Principles and Minimal Surfaces Annali Della Scuola Normale Superiore Di Pisa, Classe Di Scienze 4E Série, Tome 25, No 3-4 (1997), P
    ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze MARIO MIRANDA Maximum principles and minimal surfaces Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4e série, tome 25, no 3-4 (1997), p. 667-681 <http://www.numdam.org/item?id=ASNSP_1997_4_25_3-4_667_0> © Scuola Normale Superiore, Pisa, 1997, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) Vol. XXV (1997), pp. 667-681667 1 Maximum Principles and Minimal Surfaces MARIO MIRANDA In memory of Ennio De Giorgi Abstract. It is well known the close connection between the classical Plateau problem and Dirichlet problem for the minimal surface equation. Paradoxically in higher dimensions that connection is even stronger, due to the existence of singular solutions for Plateau problem. This fact was emphasized by Fleming’s remark [15] about the existence of such singular solutions, as a consequence of the existence of non-trivial entire solutions for the minimal surface equation. The main goal of this article is to show how generalized solutions [26] apply to the study of both, singular and regular minimal surfaces, with particular emphasis on Dirichlet and Bernstein problems, and the problem of removable singularities.
    [Show full text]
  • Minimal Surfaces
    Minimal Surfaces Connor Mooney 1 Contents 1 Introduction 3 2 BV Functions 4 2.1 Caccioppoli Sets . 4 2.2 Bounded Variation . 5 2.3 Approximation by Smooth Functions and Applications . 6 2.4 The Coarea Formula and Applications . 8 2.5 Traces of BV Functions . 9 3 Regularity of Caccioppoli Sets 13 3.1 Measure Theoretic vs Topological Boundary . 13 3.2 The Reduced Boundary . 13 3.3 Uniform Density Estimates . 14 3.4 Blowup of the Reduced Boundary . 15 4 Existence and Compactness of Minimal Surfaces 18 5 Uniform Density Estimates 20 6 Monotonicity Formulae 22 6.1 Monotonicity for Minimal Surfaces . 22 6.2 Blowup Limits of Minimal Surfaces . 25 7 Improvement of Flatness 26 7.1 ABP Estimate . 26 7.2 Harnack Inequality . 27 7.3 Improvement of Flatness . 28 8 Minimal Cones 30 8.1 Energy Gap . 30 8.2 First and Second Variation of Area . 30 8.3 Bernstein-Type Results . 31 8.4 Singular Set . 32 8.5 The Simons Cone . 33 9 Appendix 36 9.1 Whitney Extension Theorem . 36 9.2 Structure Theorems for Caccioppoli Sets . 37 9.3 Monotonicity Formulae for Harmonic Functions . 39 2 1 Introduction These notes outline De Giorgi's theory of minimal surfaces. They are based on part of a course given by Ovidiu Savin during Fall 2011. The sections on BV functions, and the regularity of Caccioppoli sets mostly follow Giusti [2] and Evans-Gariepy [1]. We give a small perturbations proof of De Giorgi's \improvement of flatness” result, a technique pioneered by Savin in [3], among other papers of his.
    [Show full text]
  • Anisotropic Energies in Geometric Measure Theory
    A NISOTROPIC E NERGIESIN G EOMETRIC M EASURE T HEORY Dissertation zur Erlangung der naturwissenschaftlichen Doktorwürde (Dr. sc. nat.) vorgelegt der Mathematisch-naturwissenschaftlichen Fakultät der Universität Zürich von antonio de rosa aus Italien Promotionskommission Prof. Dr. Camillo De Lellis (Vorsitz) Prof. Dr. Guido De Philippis Prof. Dr. Thomas Kappeler Prof. Dr. Benjamin Schlein Zürich, 2017 ABSTRACT In this thesis we focus on different problems in the Calculus of Variations and Geometric Measure Theory, with the common peculiarity of dealing with anisotropic energies. We can group them in two big topics: 1. The anisotropic Plateau problem: Recently in [37], De Lellis, Maggi and Ghiraldin have proposed a direct approach to the isotropic Plateau problem in codimension one, based on the “elementary” theory of Radon measures and on a deep result of Preiss concerning rectifiable measures. In the joint works [44],[38],[43] we extend the results of [37] respectively to any codimension, to the anisotropic setting in codimension one and to the anisotropic setting in any codimension. For the latter result, we exploit the anisotropic counterpart of Allard’s rectifiability Theorem, [2], which we prove in [42]. It asserts that every d-varifold in Rn with locally bounded anisotropic first variation is d-rectifiable when restricted to the set of points in Rn with positive lower d-dimensional density. In particular we identify a necessary and sufficient condition on the Lagrangian for the validity of the Allard type rectifiability result. We are also able to prove that in codimension one this condition is equivalent to the strict convexity of the integrand with respect to the tangent plane.
    [Show full text]
  • Conformal and Cr Mappings on Carnot Groups
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, SERIES B Volume 7, Pages 67–81 (June 17, 2020) https://doi.org/10.1090/bproc/48 CONFORMAL AND CR MAPPINGS ON CARNOT GROUPS MICHAEL G. COWLING, JI LI, ALESSANDRO OTTAZZI, AND QINGYAN WU (Communicated by Jeremy Tyson) Abstract. We consider a class of stratified groups with a CR structure and a compatible control distance. For these Lie groups we show that the space of conformal maps coincide with the space of CR and anti-CR diffeomorphisms. Furthermore, we prove that on products of such groups, all CR and anti-CR maps are product maps, up to a permutation isomorphism, and affine in each component. As examples, we consider free groups on two generators, and show that these admit very simple polynomial embeddings in CN that induce their CR structure. 1. Introduction In this article, we consider the interplay between metric and complex geometry on some model manifolds. This is the first outcome of a larger project which aims to develop a unified theory of conformal and CR structures on the one hand, and to define explicit polynomial embeddings of certain CR manifolds into Cn on the other. This kind of embedding is what permits the detailed analysis of the ∂b operator carried out by Stein and his collaborators since the 1970s (see [6]). The analogy between CR and conformal geometry is well documented in the case of CR manifolds of hypersurface-type, see, e.g., [7,10–14,17]. The easiest and perhaps the most studied example in this setting is that of the Heisenberg group, taken with its sub-Riemannian structure.
    [Show full text]
  • Arxiv:1604.08579V1 [Math.MG] 28 Apr 2016 ..Caatrztoso I Groups Lie of Characterizations 5.1
    A PRIMER ON CARNOT GROUPS: HOMOGENOUS GROUPS, CC SPACES, AND REGULARITY OF THEIR ISOMETRIES ENRICO LE DONNE Abstract. Carnot groups are distinguished spaces that are rich of structure: they are those Lie groups equipped with a path distance that is invariant by left-translations of the group and admit automorphisms that are dilations with respect to the distance. We present the basic theory of Carnot groups together with several remarks. We consider them as special cases of graded groups and as homogeneous metric spaces. We discuss the regularity of isometries in the general case of Carnot-Carathéodory spaces and of nilpotent metric Lie groups. Contents 0. Prototypical examples 4 1. Stratifications 6 1.1. Definitions 6 1.2. Uniqueness of stratifications 10 2. Metric groups 11 2.1. Abelian groups 12 2.2. Heisenberg group 12 2.3. Carnot groups 12 2.4. Carnot-Carathéodory spaces 12 2.5. Continuity of homogeneous distances 13 3. Limits of Riemannian manifolds 13 arXiv:1604.08579v1 [math.MG] 28 Apr 2016 3.1. Tangents to CC-spaces: Mitchell Theorem 14 3.2. Asymptotic cones of nilmanifolds 14 4. Isometrically homogeneous geodesic manifolds 15 4.1. Homogeneous metric spaces 15 4.2. Berestovskii’s characterization 15 5. A metric characterization of Carnot groups 16 5.1. Characterizations of Lie groups 16 Date: April 29, 2016. 2010 Mathematics Subject Classification. 53C17, 43A80, 22E25, 22F30, 14M17. This essay is a survey written for a summer school on metric spaces. 1 2 ENRICO LE DONNE 5.2. A metric characterization of Carnot groups 17 6. Isometries of metric groups 17 6.1.
    [Show full text]