Curvature-dimension Condition Research Articles

A quantitative second order estimate for (weighted) p-harmonic functions in manifolds under curvature-dimension condition

We build up a quantitative second-order Sobolev estimate of ln⁡w for positive p-harmonic functions w in Riemannian manifolds under Ricci curvature bounded from below and also for positive weighted p-harmonic functions w in weighted manifolds under the Bakry-Émery curvature-dimension condition.

Li-Yau and Harnack inequalities via curvature-dimension conditions for discrete long-range jump operators including the fractional discrete Laplacian

Li-Yau and Harnack inequalities via curvature-dimension conditions for discrete long-range jump operators including the fractional discrete Laplacian

Timelike Ricci bounds for low regularity spacetimes by optimal transport

We prove that a globally hyperbolic smooth spacetime endowed with a [Formula: see text]-Lorentzian metric whose Ricci tensor is bounded from below in all timelike directions, in a distributional sense, obeys the timelike measure-contraction property. This result includes a class of spacetimes with borderline regularity for which local existence results for the vacuum Einstein equation are known in the setting of spaces with timelike Ricci bounds in a synthetic sense. In particular, these spacetimes satisfy timelike Brunn–Minkowski, Bonnet–Myers, and Bishop–Gromov inequalities in sharp form, without any timelike nonbranching assumption. If the metric is even [Formula: see text], in fact the stronger timelike curvature-dimension condition holds. In this regularity, we also obtain uniqueness of chronological optimal couplings and chronological geodesics.

Aronson–Bénilan and Harnack estimates for the discrete porous medium equation

We consider the porous medium equation (PME) on a locally finite graph and identify suitable curvature-dimension (CD) conditions under which a discrete version of the fundamental Aronson–Bénilan estimate holds true for positive solutions of the PME on finite graphs. We also show that these estimates allow to prove Harnack inequalities which are structurally similar to the continuous case. The new CD conditions are illustrated with several concrete examples, e.g. complete and chain-like graphs.

Stability of Curvature-Dimension Condition for Negative Dimensions Under Concentration Topology

In this paper, we prove the stability of metric measure spaces satisfying the curvature-dimension condition for negative dimensions under the concentration topology. This result is an analog of the result by Funano–Shioya with respect to the dimension parameter.

Convergence of metric measure spaces satisfying the CD condition for negative values of the dimension parameter

We study the problem of whether the curvature-dimension condition with negative values of the generalized dimension parameter is stable under a suitable notion of convergence. To this purpose, first of all we propose an appropriate setting to introduce the CD(K,N) condition for N<0, allowing metric measure structures in which the reference measure is quasi-Radon. Then in this class of spaces we define the distance diKRW, which extends the already existing notions of distance between metric measure spaces. Finally, we prove that if a sequence of metric measure spaces satisfying the CD(K,N) condition with N<0 is converging with respect to the distance diKRW to some metric measure space, then this limit structure is still a CD(K,N) space.

Failure of curvature-dimension conditions on sub-Riemannian manifolds via tangent isometries

We prove that, on any sub-Riemannian manifold endowed with a positive smooth measure, the Bakry–Émery inequality for the corresponding sub-Laplacian,12Δ(‖∇u‖2)≥g(∇u,∇Δu)+K‖∇u‖2,K∈R, implies the existence of enough Killing vector fields on the tangent cone to force the latter to be Euclidean at each point, yielding the failure of the curvature-dimension condition in full generality. Our approach does not apply to non-strictly-positive measures. In fact, we prove that the weighted Grushin plane does not satisfy any curvature-dimension condition, but, nevertheless, does admit an a.e. pointwise version of the Bakry–Émery inequality. As recently observed by Pan and Montgomery, one half of the weighted Grushin plane satisfies the RCD(0,N) condition, yielding a counterexample to gluing theorems in the RCD setting.

Rényi's entropy on Lorentzian spaces. Timelike curvature-dimension conditions

For a Lorentzian space measured by m in the sense of Kunzinger, Sämann, Cavalletti, and Mondino, we introduce and study synthetic notions of timelike lower Ricci curvature bounds by K∈R and upper dimension bounds by N∈[1,∞), namely the timelike curvature-dimension conditions TCDp(K,N) and TCDp⁎(K,N) in weak and strong forms, where p∈(0,1), and the timelike measure-contraction properties TMCP(K,N) and TMCP⁎(K,N). These are formulated by convexity properties of the Rényi entropy with respect to m along ℓp-geodesics of probability measures.We show many features of these notions, including their compatibility with the smooth setting, sharp geometric inequalities, stability, equivalence of the named weak and strong versions, local-to-global properties, and uniqueness of chronological ℓp-optimal couplings and chronological ℓp-geodesics. We also prove the equivalence of TCDp⁎(K,N) and TMCP⁎(K,N) to their respective entropic counterparts in the sense of Cavalletti and Mondino.Some of these results are obtained under timelike p-essential nonbranching, a concept which is a priori weaker than timelike nonbranching.

Removable sets and [formula omitted]-uniqueness on manifolds and metric measure spaces

We study symmetric diffusion operators on metric measure spaces. Our main question is whether essential self-adjointness or Lp-uniqueness are preserved under the removal of a small closed set from the space. We provide characterizations of the critical size of removed sets in terms of capacities and Hausdorff dimension without any further assumption on removed sets. As a key tool we prove a non-linear truncation result for potentials of nonnegative functions. Our results are robust enough to be applied to Laplace operators on general Riemannian manifolds as well as sub-Riemannian manifolds and metric measure spaces satisfying curvature-dimension conditions. For non-collapsing Ricci limit spaces with two-sided Ricci curvature bounds we observe that the self-adjoint Laplacian is already fully determined by the classical Laplacian on the regular part.

Ricci Curvature on Birth-Death Processes

In this paper, we study curvature dimension conditions on birth-death processes which correspond to linear graphs, i.e., weighted graphs supported on the infinite line or the half line. We give a combinatorial characterization of Bakry and Émery’s CD(K,n) condition for linear graphs and prove the triviality of edge weights for every linear graph supported on the infinite line Z with non-negative curvature. Moreover, we show that linear graphs with curvature decaying not faster than −R2 are stochastically complete. We deduce a type of Bishop-Gromov comparison theorem for normalized linear graphs. For normalized linear graphs with non-negative curvature, we obtain the volume doubling property and the Poincaré inequality, which yield Gaussian heat kernel estimates and parabolic Harnack inequality by Delmotte’s result. As applications, we generalize the volume growth and stochastic completeness properties to weakly spherically symmetric graphs. Furthermore, we give examples of infinite graphs with a positive lower curvature bound.

Sobolev’s inequality under a curvature-dimension condition

In this note we present a new proof of Sobolev’s inequality under a uniform lower bound of the Ricci curvature. This result was initially obtained in 1983 by Ilias. Our goal is to present a very short proof, to give a review of the famous inequality and to explain how our method, relying on a gradient-flow interpretation, is simple and robust. In particular, we elucidate computations used in numerous previous works, starting with Bidaut–Véron and Véron’s 1991 classical work.

Almost-Riemannian manifolds do not satisfy the curvature-dimension condition

The Lott–Sturm–Villani curvature-dimension condition textsf{CD}(K,N) provides a synthetic notion for a metric measure space to have curvature bounded from below by K and dimension bounded from above by N. It was proved by Juillet (Rev Mat Iberoam 37(1), 177–188, 2021) that a large class of sub-Riemannian manifolds do not satisfy the textsf{CD}(K,N) condition, for any Kin {mathbb {R}} and Nin (1,infty ). However, his result does not cover the case of almost-Riemannian manifolds. In this paper, we address the problem of disproving the textsf{CD} condition in this setting, providing a new strategy which allows us to contradict the one-dimensional version of the textsf{CD} condition. In particular, we prove that 2-dimensional almost-Riemannian manifolds and strongly regular almost-Riemannian manifolds do not satisfy the textsf{CD}(K,N) condition for any Kin {mathbb {R}} and Nin (1,infty ).

Good geodesics satisfying the timelike curvature-dimension condition

Let (M,d,m,≪,≤,τ) be a causally closed, K-globally hyperbolic, regular measured Lorentzian geodesic space satisfying the weak timelike curvature-dimension condition wTCDpe(K,N) in the sense of Cavalletti and Mondino. We prove the existence of geodesics of probability measures on M which satisfy the entropic semiconvexity inequality defining wTCDpe(K,N) and whose densities with respect to m are additionally uniformly L∞ in time. This holds apart from any nonbranching assumption. We also discuss similar results under the timelike measure-contraction property.

Causal bubbles in globally hyperbolic spacetimes

We give an example of a spacetime with a continuous metric which is globally hyperbolic and exhibits causal bubbling. The metric moreover splits orthogonally into a timelike and a spacelike part. We discuss our example in the context of energy conditions and the recently introduced synthetic timelike curvature-dimension (TCD) condition. In particular we observe that the TCD-condition does not, by itself, prevent causal bubbling.

A first-order condition for the independence on p of weak gradients

It is well known that on arbitrary metric measure spaces, the notion of minimal p-weak upper gradient may depend on p.In this paper we investigate how a first-order condition of the metric-measure structure, that we call Bounded Interpolation Property, guarantees that in fact such dependence is not present.We also show that the Bounded Interpolation Property is stable for pointed measure Gromov Hausdorff convergence and holds on a large class of spaces satisfying curvature dimension conditions.

Curvature-Dimension Conditions for Symmetric Quantum Markov Semigroups

Following up on the recent work on lower Ricci curvature bounds for quantum systems, we introduce two noncommutative versions of curvature-dimension bounds for symmetric quantum Markov semigroups over matrix algebras. Under suitable such curvature-dimension conditions, we prove a family of dimension-dependent functional inequalities, a version of the Bonnet–Myers theorem and concavity of entropy power in the noncommutative setting. We also provide examples satisfying certain curvature-dimension conditions, including Schur multipliers over matrix algebras, Herz–Schur multipliers over group algebras and generalized depolarizing semigroups.

Tamed spaces – Dirichlet spaces with distribution-valued Ricci bounds

We develop the theory of tamed spaces which are Dirichlet spaces with distribution-valued lower bounds on the Ricci curvature and investigate these from an Eulerian point of view. To this end we analyze in detail singular perturbations of Dirichlet form by a broad class of distributions. The distributional Ricci bound is then formulated in terms of an integrated version of the Bochner inequality using the perturbed energy form and generalizing the well-known Bakry-Émery curvature-dimension condition. Among other things we show the equivalence of distributional Ricci bounds to gradient estimates for the heat semigroup in terms of the Feynman-Kac semigroup induced by the taming distribution as well as consequences in terms of functional inequalities. We give many examples of tamed spaces including in particular Riemannian manifolds with interior singularities and singular boundary behavior.

Long-time Behaviour of Entropic Interpolations

In this article we investigate entropic interpolations. These measure valued curves describe the optimal solutions of the Schrödinger problem Schrödinger (Sitzungsberichte Preuss Akad. Wiss. Berlin. Phys. Math. 144:144–153 1931), which is the problem of finding the most likely evolution of a system of independent Brownian particles conditionally to observations. It is well known that in the short time limit entropic interpolations converge to the McCann-geodesics of optimal transport. Here we focus on the long-time behaviour, proving in particular asymptotic results for the entropic cost and establishing the convergence of entropic interpolations towards the heat equation, which is the gradient flow of the entropy according to the Otto calculus interpretation. Explicit rates are also given assuming the Bakry-Émery curvature-dimension condition. In this respect, one of the main novelties of our work is that we are able to control the long time behavior of entropic interpolations assuming the CD(0,n) condition only.

Convergence of energy functionals and stability of lower bounds of Ricci curvature via metric measure foliation

The notion of the metric measure foliation is introduced by Galaz-Garcia, Kell, Mondino, and Sosa. They studied the relation between a metric measure space with a metric measure foliation and its quotient space. They showed that the curvature-dimension condition and the Cheeger energy functional preserve from a such space to its quotient space. Via the metric measure foliation, we investigate the convergence theory for a sequence of metric measure spaces whose dimensions are unbounded.

Porous media equation on locally finite graphs

In this paper, we consider two typical problems on a locally finite connected graph. The first one is to study the Bochner formula for the Laplacian operator on a locally finite connected graph. The other one is to obtain global nontrivial nonnegative solution to porous-media equation via the use of Aronson-Benilan argument. We use the curvature dimension condition to give a characterization two point graph. We also give a porous-media equation criterion about stochastic completeness of the graph. There is not much work in the direction of the study of nonlinear heat equations on locally finite connected graphs.