Curvature-Dimension Bounds and Functional Inequalities : Localization, Tensorization and Stability

This work is devoted to the analysis of abstract metric measure spaces (M,d,m) satisfying the curvature-dimension condition CD(K,N) presented by Sturm and in a similar form by Lott and Villani. In the first part, we introduce the notion of a Borell-Brascamp-Lieb inequality in the setting of metric measure spaces denoted by BBL(K,N). This inequality holds true on metric measure spaces fulfilling the curvature-dimension condition CD(K,N) and is stable under convergence of metric measure spaces with respect to the transportation distance. In the second part, we prove that the local version of CD(K,N) is equivalent to a global condition CD*(K,N), slightly weaker than the usual global one. This so-called reduced curvature-dimension condition CD*(K,N) has the localization property. Furthermore, we show its stability and the tensorization property. As an application we conclude that the fundamental group of a metric measure space (M,d,m) is finite whenever it satisfies locally the curvature-dimension condition CD(K,N) with positive K and finite N. In the third part, we study cones over metric measure spaces. We deduce that the n-Euclidean cone over an n-dimensional Riemannian manifold whose Ricci curvature is bounded from below by n-1 satisfies the curvature-dimension condition CD(0,n+1) and that the n-spherical cone over the same manifold fulfills CD(n,n+1)

A new construction of anticode-optimal Grassmannian codes

In this paper, we consider the well-known unital embedding from \FF_{q^k} into M_k(\FF_q) seen as a map of vector spaces over \FF_q and apply this map in a linear block code of rate $\rho/\ell$ over \FF_{q^k}. This natural extension gives rise to a rank-metric code with $k$ rows, $k\ell$ columns, dimension $\rho$ and minimum distance $k$ that satisfies the Singleton bound. Given a specific skeleton code, this rank-metric code can be seen as a Ferrers diagram rank-metric code by appending zeros on the left side so that it has length $n-k$. The generalized lift of this Ferrers diagram rank-metric code is a Grassmannian code. By taking the union of a family of the generalized lift of Ferrers diagram rank-metric codes, a Grassmannian code with length $n$, cardinality $\frac{q^n-1}{q^k-1}$, minimum injection distance $k$ and dimension $k$ that satisfies the anticode upper bound can be constructed

Geometric singularities and a flow tangent to the Ricci flow

We consider a geometric flow introduced by Gigli and Mantegazza which, in the case of smooth compact manifolds with smooth metrics, is tangen- tial to the Ricci flow almost-everywhere along geodesics. To study spaces with geometric singularities, we consider this flow in the context of smooth manifolds with rough metrics with sufficiently regular heat kernels. On an appropriate non- singular open region, we provide a family of metric tensors evolving in time and provide a regularity theory for this flow in terms of the regularity of the heat kernel. When the rough metric induces a metric measure space satisfying a Riemannian Curvature Dimension condition, we demonstrate that the distance induced by the flow is identical to the evolving distance metric defined by Gigli and Mantegazza on appropriate admissible points. Consequently, we demonstrate that a smooth compact manifold with a finite number of geometric conical singularities remains a smooth manifold with a smooth metric away from the cone points for all future times. Moreover, we show that the distance induced by the evolving metric tensor agrees with the flow of RCD(K, N) spaces defined by Gigli-Mantegazza.Comment: Fixed proof of Lemma 5.4, updated references to published work