Equality in Borell-Brascamp-Lieb inequalities on curved spaces

By using optimal mass transportation and a quantitative H\"older inequality, we provide estimates for the Borell-Brascamp-Lieb deficit on complete Riemannian manifolds. Accordingly, equality cases in Borell-Brascamp-Lieb inequalities (including Brunn-Minkowski and Pr\'ekopa-Leindler inequalities) are characterized in terms of the optimal transport map between suitable marginal probability measures. These results provide several qualitative applications both in the flat and non-flat frameworks. In particular, by using Caffarelli's regularity result for the Monge-Amp\`ere equation, we {give a new proof} of Dubuc's characterization of the equality in Borell-Brascamp-Lieb inequalities in the Euclidean setting. When the $n$-dimensional Riemannian manifold has Ricci curvature ${\rm Ric}(M)\geq (n-1)k$ for some $k\in \mathbb R$, it turns out that equality in the Borell-Brascamp-Lieb inequality is expected only when a particular region of the manifold between the marginal supports has constant sectional curvature $k$. A precise characterization is provided for the equality in the Lott-Sturm-Villani-type distorted Brunn-Minkowski inequality on Riemannian manifolds. Related results for (not necessarily reversible) Finsler manifolds are also presented.Comment: 28 pages (with 1 figure); to appear in Advances in Mathematic

Weak contact equations for mappings into Heisenberg groups

Let k>n be positive integers. We consider mappings from a subset of k-dimensional Euclidean space R^k to the Heisenberg group H^n with a variety of metric properties, each of which imply that the mapping in question satisfies some weak form of the contact equation arising from the sub-Riemannian structure of the Heisenberg group. We illustrate a new geometric technique that shows directly how the weak contact equation greatly restricts the behavior of the mappings. In particular, we provide a new and elementary proof of the fact that the Heisenberg group H^n is purely k-unrectifiable. We also prove that for an open set U in R^k, the rank of the weak derivative of a weakly contact mapping in the Sobolev space W^{1,1}_{loc}(U;R^{2n+1}) is bounded by $n$ almost everywhere, answering a question of Magnani. Finally we prove that if a mapping from U to H^n is s-H\"older continuous, s>1/2, and locally Lipschitz when considered as a mapping into R^{2n+1}, then the mapping cannot be injective. This result is related to a conjecture of Gromov.Comment: 28 page

Geometric inequalities on Heisenberg groups

We establish geometric inequalities in the sub-Riemannian setting of the Heisenberg group $\mathbb H^n$. 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 Schmuckenschl\"ager. The latter statement implies sub-Riemannian versions of the geodesic Pr\'ekopa-Leindler and Brunn-Minkowski inequalities. The proofs are based on optimal mass transportation and Riemannian approximation of $\mathbb H^n$ 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.Comment: to appear in Calculus of Variations and Partial Differential Equations (42 pages, 1 figure

Quasiconformal mappings that highly distort dimensions of many parallel lines

We construct a quasiconformal mapping of $n$-dimensional Euclidean space, $n \geq 2$, that simultaneously distorts the Hausdorff dimension of a nearly maximal collection of parallel lines by a given amount. This answers a question of Balogh, Monti, and Tyson.Comment: 12 page

Projection theorems in hyperbolic space

We establish Marstrand-type projection theorems for orthogonal projections along geodesics onto m-dimensional subspaces of hyperbolic $n$-space by a geometric argument. Moreover, we obtain a Besicovitch-Federer type characterization of purely unrectifiable sets in terms of these hyperbolic orthogonal projections.Comment: 6 pages, 2 figure

Sharp geometric inequalities in spaces with nonnegative Ricci curvature and Euclidean volume growth

By the method of optimal mass transport we prove a sharp isoperimetric inequality in ${\sf CD} (0,N)$ metric measure spaces involving the asymptotic volume ratio at infinity, $N>1$. In the setting of $n$-dimensional Riemannian manifolds with nonnegative Ricci curvature we provide a rigidity result stating that isoperimetric sets with smooth regular boundary exist if and only if the manifold is isometric to the Euclidean space. As applications of the isoperimetric inequalities, we establish Gagliardo-Nirenberg and Rayleigh-Faber-Krahn inequalities with explicit sharp constants in Riemannian manifolds with nonnegative Ricci curvature; here we use appropriate symmetrization techniques and optimal volume non-collapsing properties. The equality cases in the latter inequalities are also characterized by stating that sufficiently smooth, nonzero extremal functions exist if and only if the Riemannian manifold is isometric to the Euclidean space.Comment: 38 pages; the paper has been considerably improved by proving sharp isoperimetric inequalities on CD(0,N) spaces and characterizing the equality case in the Riemannian settin

Geometric characterizations of Gromov hyperbolicity

We prove the equivalence of three different geometric properties of metric-measure spaces with controlled geometry. The first property is the Gromov hyperbolicity of the quasihyperbolic metric. The second is a slice condition and the third is a combination of the Gehring-Hayman property and a separation conditio

Hausdorff Dimensions of Self-Similar and Self-Affine Fractals in the Heisenberg Group

We study the Hausdorff dimensions of invariant sets for self-similar and self-affine iterated function systems in the Heisenberg group. In our principal result we obtain almost sure formulae for the dimensions of self-affine invariant sets, extending to the Heisenberg setting some results of Falconer and Solomyak in Euclidean space. As an application, we complete the proof of the comparison theorem for Euclidean and Heisenberg Hausdorff dimension initiated by Balogh, Rickly and Serra-Cassano. 2000 Mathematics Subject Classification 22E30, 28A78 (primary), 26A18, 28A78 (secondary

Isometries and isometric embeddings of Wasserstein spaces over the Heisenberg group

Our purpose in this paper is to study isometries and isometric embeddings of the $p$-Wasserstein space $\mathcal{W}_p(\mathbb{H}^n)$ over the Heisenberg group $\mathbb{H}^n$ for all $p\geq1$ and for all $n\geq1$. First, we create a link between optimal transport maps in the Euclidean space $\mathbb{R}^{2n}$ and the Heisenberg group $\mathbb{H}^n$. Then we use this link to understand isometric embeddings of $\mathbb{R}$ and $\mathbb{R}_+$ into $\mathcal{W}_p(\mathbb{H}^n)$ for $p>1$. That is, we characterize complete geodesics and geodesic rays in the Wasserstein space. Using these results we determine the metric rank of $\mathcal{W}_p(\mathbb{H}^n)$. Namely, we show that $\mathbb{R}^k$ can be embedded isometrically into $\mathcal{W}_p(\mathbb{H}^n)$ for $p>1$ if and only if $k\leq n$. As a consequence, we conclude that $\mathcal{W}_p(\mathbb{R}^k)$ and $\mathcal{W}_p(\mathbb{H}^k)$ can be embedded isometrically into $\mathcal{W}_p(\mathbb{H}^n)$ if and only if $k\leq n$. In the second part of the paper, we study the isometry group of $\mathcal{W}_p(\mathbb{H}^n)$ for $p\geq1$. We find that these spaces are all isometrically rigid meaning that for every isometry $\Phi:\mathcal{W}_p(\mathbb{H}^n)\to\mathcal{W}_p(\mathbb{H}^n)$ there exists a $\psi:\mathbb{H}^n\to\mathbb{H}^n$ such that $\Phi=\psi_{\#}$. Although the conclusion is the same for $p=1$ and $p>1$, the proofs are completely different, as in the $p>1$ case the proof relies on a description of complete geodesics, and such a description is not available if $p=1$.Comment: 29 page

Sharp log-Sobolev inequalities in CD(0,N){\sf CD}(0,N) spaces with applications

Given $p,N>1,$ we prove the sharp $L^p$-log-Sobolev inequality on noncompact metric measure spaces satisfying the ${\sf CD}(0,N)$ condition, where the optimal constant involves the asymptotic volume ratio of the space. This proof is based on a sharp isoperimetric inequality in ${\sf CD}(0,N)$ spaces, symmetrisation, and a careful scaling argument. As an application we establish a sharp hypercontractivity estimate for the Hopf-Lax semigroup in ${\sf CD}(0,N)$ spaces. The proof of this result uses Hamilton-Jacobi inequality and Sobolev regularity properties of the Hopf-Lax semigroup, which turn out to be essential in the present setting of nonsmooth and noncompact spaces. Furthermore, a sharp Gaussian-type $L^2$-log-Sobolev inequality is also obtained in ${\sf RCD}(0,N)$ spaces. Our results are new, even in the smooth setting of Riemannian/Finsler manifolds. In particular, an extension of the celebrated rigidity result of Ni (J. Geom. Anal., 2004) on Riemannian manifolds will be a simple consequence of our sharp log-Sobolev inequality.Comment: Published in J. Funct. Ana