arXiv research

This paper is devoted to the analysis of metric measure spaces satisfying locally the curvature-dimension condition CD(K,N) introduced by the second author and also studied by Lott & Villani. 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…

We derive precise transformation formulas for synthetic lower Ricci bounds under time change. More precisely, for local Dirichlet forms we study how the curvature-dimension condition in the sense of Bakry-Emery will transform under time change. Similarly, for metric measure spaces we study how the curvature-dimension c…

Study shows failure of curvature-dimension conditions on sub-Riemannian manifolds.

New curvature-dimension condition for Lagrangians on manifolds.

Estimates for harmonic functions in curved spaces.

Paper proves Brunn-Minkowski inequality and curvature dimension condition are equivalent in weighted Riemannian manifolds.

We introduce and study the conical curvature-dimension condition, $CCD(K,N)$, for graphs. We show that $CCD(K,N)$ provides necessary and sufficient conditions for the underlying graph to satisfy a sharp global Poincaré inequality which in turn translates to a sharp lower bound for the first eigenvalues of these graphs.…

We study curvature dimension inequalities for the sub-Laplacian on contact Riemannian manifolds. This new curvature dimension condition is then used to obtain: 1) Geometric conditions ensuring the compactness of the underlying manifold (Bonnet-Myers type results); 2) Volume estimates of metric balls; 3) Gradient bounds…

New sub-Riemannian spaces with boundary meet curvature-dimension condition.

Study confirms equivalence in Heisenberg groups between curvature-dimension conditions and strong Brunn-Minkowski inequalities.

Researchers prove inequalities for reaction-diffusion systems using a new curvature-dimension condition.

The paper characterizes curvature-dimension conditions and related inequalities on Riemannian manifolds.

The curvature-dimension condition is a generalization of the Bochner inequality to weighted Riemannian manifolds and general metric measure spaces. It is now known to be equivalent to evolution variational inequalities for the heat semigroup, and quadratic Wasserstein distance contraction properties at different times.…

Study $L^p$ boundedness of Riesz transform on differential forms for certain manifolds.

Geodesics found in spacetime satisfy curvature conditions.

Paper proves curvature conditions are preserved in metric spaces.

The study establishes a curvature-dimension condition for discrete Markov chains.

We study some equivalent properties of the curvature-dimension conditions $CD(n,K)$ inequality on infinite, but locally finite graph. These equivalences are gradient estimate, Poincaré type inequalities and reverse Poincaré inequalities. And we also obtain one equivalent property of gradient estimate for a new notion o…

The main result of this article states that the (K;N)-cone over some metric measure space satisfies the reduced Riemannian curvature-dimension condition RCD^*(KN;N+1) if and only if the underlying space satisfies RCD^*(N-1;N). The proof uses a characterization of reduced Riemannian curvature-dimension bounds by Bochner…

The curvature-dimension condition fails in sub-Finsler geometry, extending previous results in sub-Riemannian geometry.

Study stability of curvature-dimension condition for negative dimensions.

We study the properties of the $n$-volumic scalar curvature in this note. Lott-Sturm-Villani's curvature-dimension condition ${\rm CD}(κ,n)$ was showed to imply Gromov's $n$-volumic scalar curvature $\geq nκ$ under an additional $n$-dimensional condition and we show the stability of $n$-volumic scalar curvature $\geq κ…

The paper examines stability of the Sobolev inequality in metric spaces with curvature dimension conditions.

Example of spacetime with causal bubbling, splitting into timelike and spacelike parts.

We prove a family of new Weitzenböck formulas on a Riemannian foliation with totally geodesic leaves. These Weitzenböck formulas are naturally parametrized by the canonical variation of the metric. As a consequence, under natural geometric conditions, the horizontal Laplacian satisfies a generalized curvature dimension…

We prove generalized lower Ricci bounds for Euclidean and spherical cones over compact Riemannian manifolds. These cones are regarded as complete metric measure spaces. We show that the Euclidean cone over an n-dimensional Riemannian manifold whose Ricci curvature is bounded from below by n-1 satisfies the curvature-di…

In this note we continue the analysis of metric measure space with variable ricci curvature bounds. First, we study $(κ,N)$-convex functions on metric spaces where $κ$ is a lower semi-continuous function, and gradient flow curves in the sense of a new evolution variational inequality that captures the information that …

The paper examines curvature-dimension bounds on sub-Finsler Heisenberg groups.

We prove a global Li-Yau inequality for a general Markov semigroup under a curvature-dimension condition. This inequality is stronger than all classical Li-Yau type inequalities known to us. On a Riemannian manifold, it is equivalent to a new parabolic Harnack inequality, both in negative and positive curvature, giving…

The paper proves a Moser-Trudinger inequality on metric spaces with curvature-dimension conditions.

The article examines entropy-information inequalities for continuous-time Markov chains under curvature-dimension conditions.

Paper investigates conditions for independence of weak gradients on metric spaces.

We introduce a new version of a curvature-dimension inequality for non-negative curvature. We use this inequality to prove a logarithmic Li-Yau inequality on finite graphs. To formulate this inequality, we introduce a non-linear variant of the calculus of Bakry and Émery. In the case of manifolds, the new calculus and …

We extend the range of $N$ to negative values in the $(K,N)$-convexity (in the sense of Erbar--Kuwada--Sturm), the weighted Ricci curvature $Ric_N$ and the curvature-dimension condition $CD(K,N)$. We generalize a number of results in the case of $N>0$ to this setting, including Bochner's inequality, the Brunn--Minkowsk…

Warped products over one-dimensional base spaces satisfy curvature-dimension condition under specific conditions.

New findings show different cost functions yield equivalent curvature bounds.

Motivated by a classical comparison result of J. C. F. Sturm we introduce a curvature-dimension condition CD(k,N) for general metric measure spaces and variable lower curvature bound k. In the case of non-zero constant lower curvature our approach coincides with the celebrated condition that was proposed by K.-T. Sturm…

We prove generalized lower Ricci bounds for Euclidean and spherical cones over complete Riemannian manifolds. These cones are regarded as complete metric measure spaces. In general, they will be neither manifolds nor Alexandrov spaces. We show that the Euclidean cone over an n-dimensional Riemannian manifold whose Ricc…

We prove the equivalence of the curvature-dimension bounds of Lott-Sturm-Villani (via entropy and optimal transport) and of Bakry--Émery (via energy and Γ_2$-calculus) in complete generality for infinitesimally Hilbertian metric measure spaces. In particular, we establish the full Bochner inequality on such metric meas…

In this note we show that in metric measure spaces satisfying the reduced curvature-dimension condition CD*(K,N) we always have geodesics in the Wasserstein space of probability measures that satisfy the critical convexity inequality of CD*(K,N) also for intermediate times and in addition the measures along these geode…

Proposes a new metric space example showing non-constant topological dimension.

In this paper, we prove the concavity of the Shannon entropy power for the heat equation associated with the Laplacian or the Witten Laplacian on complete Riemannian manifolds with suitable curvature-dimension condition and on compact super Ricci flows. Under suitable curvature-dimension condition, we prove that the ri…

Using the curvature-dimension inequality proved in Part~I, we look at consequences of this inequality in terms of the interaction between the sub-Riemannian geometry and the heat semigroup $P_t$ corresponding to the sub-Laplacian. We give bounds for the gradient, entropy, a Poincaré inequality and a Li-Yau type inequal…

We study Bakry-Emery type estimates for the Laplace-Beltrami operator of a totally geodesic foliation. In particular, we are interested in situations for which the $Γ_2$ operator may not be bounded from below but the horizontal Bakry-Emery curvature is. As we prove it, under a bracket generating condition, this weaker …

Timelike curvature and Brunn-Minkowski inequality linked in non-smooth spacetimes.

We show that the family of probability measures on the $n$-dimensional unit sphere, having density proportional to: \[ S^n \ni y \mapsto \frac{1}{|y - x|^{n+α}}, \] satisfies the Curvature-Dimension condition $CD(n-1-\frac{n+α}{4},-α)$, for all $|x| < 1$, $α\geq -n$ and $n\geq 2$. The case $α= 1$ corresponds to the hit…

Based on a study of the coupling by reflection of diffusion processes, a new monotonicity in time of a time-dependent transportation cost between heat distribution is shown under Bakry-Emery's curvature-dimension condition on a Riemannian manifold. The cost function comes from the total variation between heat distribut…

Study introduces new curvature conditions for Lorentzian spaces using Rényi entropy.