arXiv research

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 …

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

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

Paper proves Gromov's cube inequality in all dimensions.

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…

Proves inequalities on curved spaces with positive curvature.

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

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

SubRiemannian structures fail to meet Riemannian Brunn--Minkowski inequalities.

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

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…

Sharp inequality in spaces with non-negative Ricci curvature.

On an asymptotically flat manifold $M^n$ with nonnegative scalar curvature, with outer minimizing boundary $Σ$, we prove a Penrose-like inequality in dimensions $n < 8$, under suitable assumptions on the mean curvature and the scalar curvature of $Σ$.

In this paper, we define the curvature dimension inequalities CD(m, K) on finite directed graphs modifying the case of undirected graphs. As a main result, we evaluate m and K on finite directed graphs.

Proves Penrose inequality in all dimensions for specific 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.…

We give a generalized curvature-dimension inequality connecting the geometry of sub-Riemannian manifolds with the properties of its sub-Laplacian. This inequality is valid on a large class of sub-Riemannian manifolds obtained from Riemannian foliations. We give a geometric interpretation of the invariants involved in t…

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

The paper extends Liouville theorems to sub-Riemannian manifolds.

Sharp inequality for submanifolds in curved spaces.

By adapting some ideas of M. Ledoux \cite{ledoux2}, \cite{ledoux-stflour} and \cite{Led} to a sub-Riemannian framework we study Sobolev, Poincaré and isoperimetric inequalities associated to subelliptic diffusion operators that satisfy the generalized curvature dimension inequality that was introduced by F. Baudoin and…

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 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 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…

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

Optimal systolic inequality proved for manifolds with positive bi-Ricci curvature.

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…

Sharp Minkowski inequality for convex surfaces in curved spaces.

Study shows mass-capacity inequality for specific geometric manifolds.

New curvature-dimension condition for Lagrangians on manifolds.

Two inequalities for convex surfaces in electrostatics.

The paper investigates higher dimensional analogues of Burago's inequality bounding the area of a closed surface by its total curvature. We obtain sufficient conditions for hypersurfaces in 4-space that involve the Ricci curvature. We get semi-local variants of the inequality holding in any dimension that involve domai…

Extends spectral torus band inequalities for compact manifolds with scalar curvature bounds.

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

A well known question in differential geometry is to control the constant in isoperimetric inequality by intrinsic curvature conditions. In dimension 2, the constant can be controlled by the integral of the positive part of the Gaussian curvature. In this paper, we showed that on simply connected conformal flat manifol…

By studying the heat semigroup, we prove Li-Yau type estimates for bounded and positive solutions of the heat equation on graphs, under the assumption of the curvature-dimension inequality $CDE'(n,0)$, which can be consider as a notion of curvature for graphs. Furthermore, we derive that if a graph has non-negative cur…

Two-dimensional Lagrangian mean curvature equation solved with new inequality.

Paper extends Willmore inequality to manifolds with negative Ricci curvature.

We show a connection between the $CDE'$ inequality and the $CDψ$ inequality. In particular, we introduce a $CD_ψ^\varphi$ inequality as a slight generalization of $CDψ$ which turns out to be equivalent to $CDE'$ with appropriate choices of $\varphi$ and $ψ$. We use this to prove that the $CDE'$ inequality implies the c…

We obtain new sharp isoperimetric inequalities on a Riemannian manifold equipped with a probability measure, whose generalized Ricci curvature is bounded from below (possibly negatively), and generalized dimension and diameter of the convex support are bounded from above (possibly infinitely). Our inequalities are shar…

The study proves Lieb-Thirring inequalities on hyperbolic manifolds.

The paper improves inequalities for Kähler-Einstein manifolds using curvature conditions.

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 …

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

We prove a sharp logarithmic Sobolev inequality which holds for submanifolds in Euclidean space of arbitrary dimension and codimension. Like the Michael-Simon Sobolev inequality, this inequality includes a term involving the mean curvature.

In this paper, we derive Li-Yau inequality for unbounded Laplacian on complete weighted graphs with the assumption of the curvature-dimension inequality $CDE'(n,K)$, which can be regarded as a notion of curvature on graphs. Furthermore, we obtain some applications of Li-Yau inequality, including Harnack inequality, hea…

We study the isoperimetric, functional and concentration properties of $n$-dimensional weighted Riemannian manifolds satisfying the Curvature-Dimension condition, when the generalized dimension $N$ is negative, and more generally, is in the range $N \in (-\infty,1)$, extending the scope from the traditional range $N \i…

The study develops inequalities for Riemannian foliations without bundle-like assumptions.