Geometric inequalities on Heisenberg groups
We establish geometric inequalities in the sub-Riemannian setting of the Heisenberg group . Our results include a natural sub-Riemannian version of the celebrated curvature-dimension condition of Lott-Villani and Sturm and also a geodesic version of the Borell-Brascamp-Lieb inequality akin to the one obtained by Cordero-Erausquin, McCann and Schmuckenschläger. The latter statement implies sub-Riemannian versions of the geodesic Prékopa-Leindler and Brunn-Minkowski inequalities. The proofs are based on optimal mass transportation and Riemannian approximation of developed by Ambrosio and Rigot. These results refute a general point of view, according to which no geometric inequalities can be derived by optimal mass transportation on singular spaces.
Dedicated to Hans Martin Reimann on the occasion of his 75th birthday
and to Cristian Gutiérrez on the occasion of his 65th birthday.
- 1 Introduction and main results
- 2 Preliminary results
- 3 Proof of main results
- 4 Geometric aspects of Brunn-Minkowski inequalities on
- 5 Concluding remarks and further questions
Keywords: Heisenberg group; curvature-dimension condition; entropy convexity; Borell-Brascamp-Lieb inequality; Prékopa-Leindler inequality; Brunn-Minkowski inequality.
MSC: 49Q20, 53C17.
1. Introduction and main results
1.1. General background and motivation
Due to the seminal papers by Lott and Villani  and Sturm [38, 39], metric measure spaces with generalized lower Ricci curvature bounds support various geometric and functional inequalities including Borell-Brascamp-Lieb, Brunn-Minkowski, Bishop-Gromov inequalities. A basic assumption for these results is the famous curvature-dimension condition which –in the case of a Riemannian manifold –, represents the lower bound for the Ricci curvature on and the upper bound for the dimension of , respectively. It is a fundamental question whether the method used in , [38, 39], based on optimal mass transportation works in the setting of singular spaces with no apriori lower curvature bounds. A large class of such spaces are the sub-Riemannian geometric structures or Carnot-Carathéodory geometries, see Gromov .
During the last decade considerable effort has been made to establish geometric and functional inequalities on sub-Riemannian spaces. The quest for Borell-Brascamp-Lieb and Brunn-Minkowski type inequalities became a hard nut to crack even on simplest sub-Riemannian setting such as the Heisenberg group endowed with the usual Carnot-Carathéodory metric and -measure. One of the reasons for this is that although there is a good first order Riemannian approximation (in the pointed Gromov-Hausdorff sense) of the sub-Riemannian metric structure of the Heisenberg group , there is no uniform lower bound on the Ricci curvature in these approximations (see e.g. Capogna, Danielli, Pauls and Tyson [11, Section 2.4.2]); indeed, at every point of there is a Ricci curvature whose limit is in the Riemannian approximation. The lack of uniform lower Ricci bounds prevents a straightforward extension of the Riemannian Borell-Brascamp-Lieb and Brunn-Minkowski inequalities of Cordero-Erausquin, McCann and Schmuckenschläger  to the setting of the Heisenberg group. Another serious warning is attributed to Juillet  who proved that both the Brunn-Minkowski inequality and the curvature-dimension condition fail on for every choice of and .
These facts tacitly established the view according to which there are no entropy-convexity and Borell-Brascamp-Lieb type inequalities on singular spaces such as the Heisenberg groups. The purpose of this paper is to deny this paradigm. Indeed, we show that the method of optimal mass transportation is powerful enough to yield good results even in the absence of lower curvature bounds. By using convergence results for optimal transport maps in the Riemannian approximation of due to Ambrosio and Rigot  we are able to introduce the correct sub-Riemannian geometric quantities which can replace the lower curvature bounds and can be successfully used to establish geodesic Borell-Brascamp-Lieb, Prékopa-Leindler, Brunn-Minkowski and entropy inequalities on the Heisenberg group . The main statements from the papers of Figalli and Juillet  and Juillet  will appear as special cases of our results.
Before stating our results we shortly recall the aforementioned geometric inequalities of Borell-Brascamp-Lieb and the curvature dimension condition of Lott-Sturm-Villani and indicate their behavior in the sub-Riemannian setting of Heisenberg groups.
1.2. An overview of geometric inequalities
The classical Borell-Brascamp-Lieb inequality in states that for any fixed , and integrable functions which satisfy
Here and in the sequel, for every , and , we consider the -mean
with the conventions , and and if and if The Borell-Brascamp-Lieb inequality reduces to the Prékopa-Leindler inequality for , which in turn implies the Brunn-Minkowski inequality
where and are positive and finite measure subsets of , and denotes the -dimensional Lebesgue measure. For a comprehensive survey on geometric inequalities in and their applications to isoperimetric problems, sharp Sobolev inequalities and convex geometry, we refer to Gardner .
In his Ph.D. Thesis, McCann [29, Appendix D] (see also ) presented an optimal mass transportation approach to Prékopa-Leindler, Brunn-Minkowski and Brascamp-Lieb inequalities in the Euclidean setting. This pioneering idea led to the extension of a geodesic version of the Borell-Brascamp-Lieb inequality on complete Riemannian manifolds via optimal mass transportation, established by Cordero-Erausquin, McCann and Schmuckenschläger . Closely related to the Borell-Brascamp-Lieb inequalities on Riemannian manifolds is the convexity of the entropy functional . The latter fact served as the starting point of the work of Lott and Villani  and Sturm [38, 39] who initiated independently the synthetic study of Ricci curvature on metric measure spaces by introducing the curvature-dimension condition for and . Their approach is based on the effect of the curvature of the space encoded in the reference distortion coefficients
where is the density function of w.r.t. and The metric measure space satisfies the curvature-dimension condition for and if and only if for every there exists an optimal coupling of and and a geodesic joining and such that for all and ,
It turns out that a Riemannian (resp. Finsler) manifold satisfies the condition if and only if the Ricci curvature on is not smaller than and the dimension of is not greater than , where is the natural metric on and m is the canonical Riemannian (resp. Busemann-Hausdorff) measure on see Sturm  and Ohta .
Coming back to the Borell-Brascamp-Lieb inequality in curved spaces, e.g., when is a complete -dimensional Riemannian manifold, we have to replace the convex combination in (1.1) by the set of -intermediate points between and w.r.t. the Riemannian metric on defined by
With this notation, we can state the result of Cordero-Erausquin, McCann and Schmuckenschläger  (see also Bacher ), as the Borell-Brascamp-Lieb inequality on which holds if and only if for all , and integrable functions satisfying
where We would like to emphasize the fact that in  the main ingredient is provided by a weighted Jacobian determinant inequality satisfied by the optimal transport interpolant map.
It turns out, even in the more general setting of non-branching geodesic metric spaces, that both and imply the geodesic Brunn-Minkowski inequality , see Bacher , i.e., if is such a space, for Borel sets with and ,
Here is the set of -intermediate points between the elements of the sets and w.r.t. the metric , defined by and
As we already pointed out, Juillet  proved that the Brunn-Minkowski inequality fails on for every choice of and ; therefore, both and fail too. In fact, a closer investigation shows that the failure of these inequalities on is not surprising: indeed, the distortion coefficient is a ’pure Riemannian’ object coming from the behavior of Jacobi fields along geodesics in Riemannian space forms. More quantitatively, since certain Ricci curvatures tend to in the Riemannian approximation of the first Heisenberg group (see Capogna, Danielli, Pauls and Tyson [11, Section 2.4.2]) and for every and , some Riemannian quantities blow up and they fail to capture the subtle sub-Riemannian metric structure of the Heisenberg group. In particular, assumption (1.3) in degenerates to an impossible condition.
On the other hand, there is a positive effect in the Riemannian approximation (see [11, Section 2.4.2]) that would be unfair to conceal. It turns out namely, that the two remaining Ricci curvatures in will blow up to in the Riemannian approximation scheme. This can be interpreted as a sign of hope for a certain cancellation that could save the day at the end. This will be indeed the case: appropriate geodesic versions of Borell-Brascamp-Lieb and Brunn-Minkowski inequalities still hold on the Heisenberg group as we show in the sequel.
1.3. Statement of main results
According to Gromov , the Heisenberg group with its sub-Riemannian, or Carnot-Carathéodory metric, can be seen as the simplest prototype of a singular space. In this paper we shall use a model of that is identified with its Lie algebra via canonical exponential coordinates. At this point we just recall the bare minimum that is needed of the metric structure of in order to state our results. In the next section we present a more detailed exposition of the Heisenberg geometry, its Riemannian approximation and the connection between their optimal mass transportation maps. We denote a point in by , where , , and we identify the pair with having coordinates for all . The correspondence with its Lie algebra through the exponential coordinates induces the group law
where denotes the imaginary part of a complex number and is the Hermitian inner product. In these coordinates the neutral element of is and the inverse element of is . Note that form a real coordinate system for and the system of vector fields given as differential operators
forms a basis for the left invariant vector fields of The vectors form the basis of the horizontal bundle and we denote by the associated Carnot-Carathéodory metric.
For the parameters , the paths are length-minimizing non-constant geodesics in joining and . If then it follows that the geodesics connecting and are unique, while for the uniqueness fails. Let
and be the center of the group . The cut-locus of is . If then is well defined. Otherwise, (if ) or (if ).
In analogy to we introduce for the Heisenberg distortion coefficients defined by
The function is increasing on (cf. Lemma 2.1), in particular as ; and also:
For , we introduce the notation
If , we let with the property that . Observe, that is well defined and . If we set .
A rough comparison of the Riemannian and Heisenberg distortion coefficients is in order. First of all, both quantities and encode the effect of the curvature in geometric inequalities. Moreover, both of them depend on the dimension of the space, as indicated by the parameter in the Riemannian case and in the Heisenberg case. However, by there is an explicit dependence of the lower bound of the Ricci curvature , while in the expression of no such dependence shows up.
Let us recall that in case of the elegant proof of the Borell-Brascamp-Lieb inequality by the method of optimal mass transportation, see e.g. Villani [41, 40] is based on the concavity of defined on the set of -dimensional real symmetric positive semidefinite matrices. In a similar fashion, Cordero-Erausquin, McCann and Schmuckenschläger derive the Borell-Brascamp-Lieb inequality on Riemannian manifolds by the optimal mass transportation approach from a concavity-type property of as well, which holds for the -dimensional matrices, obtained as Jacobians of the map . Here is a -concave map defined on the complete Riemannian manifold , is the Riemannian metric, and and denote the exponential map and Riemannian gradient on . Here, the concavity is for the Jacobian matrices , where is the interpolant map defined for -a.e. as
Here is the set of -intermediate points between w.r.t. to the Riemannian metric , and is the optimal transport map between the absolutely continuous probability measures and defined on minimizing the transportation cost w.r.t. the quadratic cost function .
Our first result is an appropriate version of the Jacobian determinant inequality on the Heisenberg group. In order to formulate the precise statement we need to introduce some more notations.
Let . Hereafter, denotes the -intermediate set associated to the nonempty sets w.r.t. the Carnot-Carathéodory metric . Note that is a geodesic metric space, thus for every .
Let and be two compactly supported probability measures on that are absolutely continuous w.r.t. . According to Ambrosio and Rigot , there exists a unique optimal transport map transporting to associated to the cost function . If denotes the interpolant optimal transport map associated to , defined as
the push-forward measure is also absolutely continuous w.r.t. , see Figalli and Juillet . Note that the maps and are essentially injective thus their inverse functions and are well defined -a.e. and -a.e., respectively, see Figalli and Rifford [16, Theorem 3.7] and Figalli and Juillet [15, p. 136]. If is not in the Heisenberg cut-locus of (i.e., which happens -a.e.) and , there exists a unique ’angle’ defined by , where is the unique pair such that . If , we set . Observe that the map is Borel measurable on .
Our main result can now be stated as follows.
(Jacobian determinant inequality on ) Let and assume that and are two compactly supported, Borel probability measures, both absolutely continuous w.r.t. on . Let be the unique optimal transport map transporting to associated to the cost function and its interpolant map. Then the following Jacobian determinant inequality holds:
If , and are the density functions of the measures , and w.r.t. to , respectively, the Monge-Ampère equations
show the equivalence of (1.9) to
It turns out that a version of Theorem 1.1 holds even in the case when only is required to be absolutely continuous. In this case we consider only the first term on the right hand side of (1.11). Inequality (1.7) shows that
The first application of Theorem 1.1 is an entropy inequality. In order to formulate the result, we recall that for a function one defines the -entropy of an absolutely continuous measure w.r.t. on as
where is the density of
Our entropy inequality is stated as follows:
(General entropy inequality on ) Let and assume that and are two compactly supported, Borel probability measures, both absolutely continuous w.r.t. on with densities and respectively. Let be the unique optimal transport map transporting to associated to the cost function and its interpolant map. If is the interpolant measure between and and is a function such that and is non-increasing and convex, the following entropy inequality holds:
Various relevant choices of admissible functions will be presented in the sequel. In particular, Theorem 1.2 provides an curvature-dimension condition on the metric measure space for the choice of
Another consequence of Theorem 1.1 is the following Borell-Brascamp-Lieb inequality:
(Weighted Borell-Brascamp-Lieb inequality on ) Fix and . Let be Lebesgue integrable functions with the property that for all
Then the following inequality holds:
Consequences of Theorem 1.3 are uniformly weighted and non-weighted Borell-Brascamp-Lieb inequalities on which are stated in Corollaries 3.5 and 3.6, respectively. As particular cases we obtain Prékopa-Leindler-type inequalities on , stated in Corollaries 3.7-3.9.
Let us emphasize the difference between the Riemannian and sub-Riemannian versions of the entropy and Borell-Brascamp-Lieb inequalites. In the Riemannian case, we notice the appearance of the distance function in the expression of . The explanation of this phenomenon is that in the Riemannian case the effect of the curvature accumulates in dependence of the distance between and in a controlled way, estimated by the lower bound of the Ricci curvature. In contrast to this fact, in the sub-Riemannian framework the argument appearing in the weight is not a distance but a quantity measuring the deviation from the horizontality of the points and , respectively. Thus, in the Heisenberg case the effect of positive curvature occurs along geodesics between points that are situated in a more vertical position with respect to each other. On the other hand an effect of negative curvature is manifested between points that are in a relative ‘horizontal position’ to each other. The size of the angle measures the ’degree of verticality’ of the relative positions of and which contributes to the curvature.
The geodesic Brunn-Minkowski inequality on the Heisenberg group will be a consequence of Theorem 1.3. For two nonempty measurable sets we introduce the quantity
where the sets and are nonempty, full measure subsets of and , respectively.
(Weighted Brunn-Minkowski inequality on ) Let and and be two nonempty measurable sets of . Then the following geodesic Brunn-Minkowski inequality holds:
Here we consider the outer Lebesgue measure whenever is not measurable, and the convention for the right hand side of (1.13). The latter case may happen e.g. when indeed, in this case and
The value represents a typical Heisenberg quantity indicating a lower bound of the deviation of an essentially horizontal position of the sets and . An intuitive description of the role of weights and in (1.13) will be given in Section 4.
By Theorem 1.4 we deduce several forms of the Brunn-Minkowski inequality, see Corollary 4.2. Moreover, the weighted Brunn-Minkowski inequality implies the measure contraction property MCP on proved by Juillet [21, Theorem 2.3], see also Corollary 4.1, namely, for every , and nonempty measurable set ,
Our proofs are based on techniques of optimal mass transportation and Riemannian approximation of the sub-Riemannian structure. We use extensively the machinery developed by Cordero-Erausquin, McCann and Schmuckenschläger  on Riemannian manifolds and the results of Ambrosio and Rigot  and Juillet  on . In our approach we can avoid the blow-up of the Ricci curvature to by not considering limits of the expressions of . Instead of this, we apply the limiting procedure to the coefficients expressed in terms of volume distortions. It turns out that one can directly calculate these volume distortion coefficients in terms of Jacobians of exponential maps in the Riemannian approximation. These quantities behave in a much better way under the limit, avoiding blow-up phenomena. The calculations are based on an explicit parametrization of the Heisenberg group and the approximating Riemannian manifolds by an appropriate set of spherical coordinates that are based on a fibration of the space by geodesics.
The paper is organized as follows. In the second section we present a series of preparatory lemmata obtaining the Jacobian representations of the volume distortion coefficients in the Riemannian approximation of the Heisenberg group and we discuss their limiting behaviour. In the third section we present the proof of our main results, i.e., the Jacobian determinant inequality, various entropy inequalities and Borell-Brascamp-Lieb inequalities. The forth section is devoted to geometric aspects of the Brunn-Minkowski inequality. In the last section we indicate further perspectives related to this research. The results of this paper have been announced in .
Acknowledgements. The authors wish to express their gratitude to Luigi Ambrosio, Nicolas Juillet, Pierre Pansu, Ludovic Rifford, Séverine Rigot and Jeremy Tyson for helpful conversations on various topics related to this paper.
2. Preliminary results
2.1. Volume distortion coefficients in
The left translation by the element is given by for all One can observe that is affine, associated to a matrix with determinant 1. Therefore the Lebesgue measure of will be the Haar measure on (uniquely defined up to a positive multiplicative constant).
For define the nonisotropic dilation as Observe that for any measurable set ,
thus the homogeneity dimension of the Lebesgue measure is on .
In order to equip the Heisenberg group with the Carnot-Carathéodory metric we consider the basis of the space of the horizontal left invariant vector fields . A horizontal curve is an absolutely continuous curve for which there exist measurable functions () such that
The length of this curve is
The classical Chow-Rashewsky theorem assures that any two points from the Heisenberg group can be joined by a horizontal curve, thus it makes sense to define the distance of two points as the infimum of lengths of all horizontal curves connecting the points, i.e.,
is called the Carnot-Carathéodory metric. The left invariance and homogeneity of the vector fields are inherited by the distance , thus