On the isoperimetric problem in the Heisenberg group Hn

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Niveau: Supérieur, Doctorat, Bac+8
On the isoperimetric problem in the Heisenberg group Hn Gian Paolo Leonardi1 and Simon Masnou2 Abstract It has been recently conjectured that, in the context of the Heisenberg group Hn endowed with its Carnot-Caratheodory metric and Haar measure, the isoperimetric sets (i.e., minimizers of the H-perimeter among sets of constant Haar measure) could coincide with the solutions to a “restricted” isoperimetric problem within the class of sets having finite perimeter, smooth boundary, and cylindrical symmetry. In this paper, we derive new properties of these restricted isoperimetric sets, that we call Heisenberg bubbles. In particular, we show that their boundary has constant mean H-curvature and, quite surprisingly, that it is foliated by the family of minimal geodesics connecting two special points. In view of a possible strategy for proving that Heisenberg bubbles are actually isoperimetric among the whole class of measurable subsets of Hn, we turn our attention to the relationship between volume, perimeter and -enlargements. In particular, we prove a Brunn-Minkowski inequality with topological exponent as well as the fact that the H-perimeter of a bounded, open set F ? Hn of class C2 can be computed via a generalized Minkowski content, defined by means of any bounded set whose horizontal projection is the 2n-dimensional unit disc. Some consequences of these properties are discussed. 2000 AMS Subject Classification.

  • brunn-minkowski inequality

  • haar measure

  • any regular

  • minkowski content

  • volume

  • heisenberg group

  • carnot groups

  • isoperimetric problem

  • perimeter among all

  • has constant


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On the isoperimetric problem in the Heisenberg groupHn Gian Paolo Leonardi1 Masnouand Simon2
Abstract It has been recently conjectured that, in the context of the Heisenberg groupHnendowed with its Carnot-Caratheodory metric and Haar measure, the isoperimetric sets (i.e., minimizers of theHmpeeri-rteamong sets of constant Haar measure) could coincide with the solutions to a “restricted” isoperimetric problem within the class of sets having nite perimeter, smooth boundary, and cylindrical symmetry. In this paper, we derive new properties of these restricted isoperimetric sets, that we callHeisenberg bubbles. In particular, we show that their boundary has constant meanH-curvature and, quite surprisingly, that it is foliated by the family of minimal geodesics connecting two special points. In view of a possible strategy for proving that Heisenberg bubbles are actually isoperimetric among thewholeclass of measurable subsets of Hn, we turn our attention to the relationship between volume, perimeter andmenest.-enlarg In particular, we prove a Brunn-Minkowski inequality with topological exponent as well as the fact that theH-perimeter of a bounded, open setFHnof classC2can be computed via a generalized Minkowski content means of any bounded set whose horizontal projection, de ned by is the 2n Some-dimensional unit disc.consequences of these properties are discussed. 
2000AMSSubjectClassi cation.28A75; 22E25; 49Q20.
1 Introduction It is well-known that Euclidean balls inRnare, up to negligible sets, the unique solutions to the isoperimetric probleminRn, that is, the unique minimizers of theperimeteramong all measurable sets with samen Therefore, we say that Euclidean balls are the-dimensional Lebesgue measure. isoperimetric setsinRn. Here, we consider the isoperimetric problem in theHeisenberg groupHn, where the Euclidean geometry ofR2n+1is replaced by aniananus-biRmegeometry induced by a certain family ofhor-izontalnoothtseutydfousb-Riemannianraeytneceesevahsinowgrnatienttgactorves.Re eld spaces (and even more general metric measure spaces) from the viewpoint of the theory of BV functions and sets of nite perimeter, and, more generally, in the framework of geometric measure theory (see for instance [2, 3, 4, 14, 15, 16, 18, 23, 28]). These spaces naturally arise from di er-ent areas of mathematics and physics, such as harmonic analysis, control theory, non-holonomic mechanics [1, 5, 11, 12], and, recently, from the theory of human vision [8]. Before giving the de nition and discussing some properties ofHn, let us point out the relation-ship between the isoperimetric problem and theisoperimetric inequalities. We recall that bothRn 1DirtpaenimidototeMMeidledoliMatematici,UnievsrtiaidaPodavlzBeia,v51,3i7onavodaP13,ylatI, leonardi@dmsa.unipd.it 2uoL-iLsicaJeseuq7,18ivUns,onC.B.eirr-etereistPeurie,752-Marie-CF,50xedeCsiraP25e,ncraaLotriobar masnou@ann.jussieu.fr
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andHnbelong to the wider class ofCarnot groups, i.e., structures of the form (G,, , dc), where (G,) is a connected and simply connected Lie group,is a (family of) dilation(s) anddcis the Carnot-Caratheodorymetric concerning(see Section 2 for more precise de nitionsHn). It is known that isoperimetric inequalities of the type
C|F|QPG(F) (1.1) hold for all measurableFGwith|F|<, and for some positiveCdepending only onG[18, 30]. Here, || denotes then-dimensional Lebesgue measure (theHaar measureof the groupG'Rn),Q is thehomogeneous dimensionofGandPG(F) denotes theG respect to the with-perimeter de ned familyofhorizontalvector elds(seeSection2).Since|  |andPGare, respectively,Qand (Q 1)-homogeneous with respect to dilations, one can write (1.1) for|F|= 1 and easily obtain that the best (the largest) constantCthat can be plugged into (1.1) is exactly the in m of umPG(F) under the volume constraint|F|= 1, that is, the perimeter of any possibly existing isoperimetric set, scaled to have unit volume.
The existence of isoperimetric sets in Carnot groups has been recently proved in [22], where some general properties of those sets are also carried out: more precisely, one can show that these sets are bounded, with Alhfors-regular boundary verifying a condition of “good” geometric separation (the so-calledcondition B). Moreover, at least for Carnot groups ofstep2 and in particular for the Heisenberg groupHn, the connectedness can also be proved as a consequence of being adomain of isoperimetry. Yet a more precise characterization of isoperimetric sets in a general Carnot group is still an open (and dicult) problem. One could expect that the natural candidate isoperimetric sets inHnare the balls associated to the Carnot-Caratheodory metric, as they are the counterparts of the Euclidean balls inRn. How-ever, as shown in a recent work by Monti [26], these balls arenot the particularisoperimetric. In case of the rst Heisenberg groupH1, a reasonably good approximation of an isoperimetric set can be obtained as the output of a numerical simulation, that we have performed with Brakke’s Surface Evolver[6].Thissimulation ndsatheoreticaljusti cationinanapproximationresultofsetsof niteH t polyhedra as initial con gura- from di eren Starting-perimeter with polyhedral sets [24]. tions, the minimization of theH-perimeter at constant volume leads, up to left-translations, to a unique, apparently smooth and convex body with an evident cylindrical symmetry (see Figure 1) plus a symmetry with respect to thez-plane (recall that the points ofHncan be seen as the pairs [z, t]CnR'R2n+1nnacnoitalumisehtwhaetteanarguotse,tcour).Ofhatwe ndisaglobal minimizer instead of a local one, but surely adds credit to the natural conjecture about the sym-metries of such isoperimetric sets, which should be coherent with the symmetries ofH1: indeed, all rotations around thet-axis, as well as the map (x, y, t)7→(x, y, t), are automorphisms ofH1 (see [11]) preserving both volume andH-perimeter. Motivated by the results of our simulations and generalizing toHn,n1, we are naturally led to consider a “restricted” isoperimetric problem, that is the minimization of the ratioPH(F)/|F|Q 1 on the subclassFof setsFwhose boundary∂Fcan be decomposed as the unionS+S , whereS+=∂F∩ {t0}radial, smooth and non identically zero functionis the graph of some g(z) =f(|z|), whereasS is the symmetric ofS+with respect to thez-plane. It can be proved that this restricted isoperimetric problem admits solutions (see Theorem 3.3) that we callHeisenberg bubbles. We believe that these are the right candidates to solve the (global)
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