The curvature-dimension condition implies a new weighted scalar curvature.
problem Studying the properties of the n-volumic scalar curvature. method Using the curvature-dimension condition mCD(κ,n) and smGH-convergence. result The stability of n-volumic scalar curvature ≥κ under smGH-convergence. 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…
Paper proves Brunn-Minkowski inequality and curvature dimension condition are equivalent in weighted Riemannian manifolds.
problem Proving equivalence between Brunn-Minkowski inequality and curvature dimension condition.
method Analyzes weighted Riemannian manifolds, proving equivalence without optimal transport or differential structure.
result Brunn-Minkowski inequality and curvature dimension condition are equivalent in weighted Riemannian manifolds.
Estimates for harmonic functions in curved spaces.
problem Quantifying harmonic functions in curved spaces.
method Quantitative Sobolev estimates for p-harmonic functions in manifolds with curvature conditions. result Established a quantitative second order Sobolev estimate for p-harmonic functions. New curvature-dimension condition for Lagrangians on manifolds.
problem Establishing a curvature-dimension condition for autonomous Lagrangians.
method Generalizing Klartag's needle decomposition technique to Lagrangian setting.
result Equivalence of curvature-dimension condition to displacement convexity of entropy.
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 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…
Paper proves curvature conditions are preserved in metric spaces.
problem Preserving curvature conditions in metric spaces.
method Doubling and gluing constructions to preserve RCD(K,N) condition. result Proves RCD(K,N) condition is preserved. Study shows failure of curvature-dimension conditions on sub-Riemannian manifolds.
problem Failure of curvature-dimension conditions on sub-Riemannian manifolds.
method Proves failure of curvature-dimension conditions using tangent isometries and Killing vector fields.
result Proves failure of curvature-dimension conditions on sub-Riemannian manifolds.
We examine algebraic conditions for the sectional positivity of the Riemann curvature operator. We describe sufficient conditions for dimension n=4, and complete characterization for a dense open subset of the space of operators in dimension 4. We also briefly examine higher-dimentional curvature operators.
The paper explores rigidity of hypersurfaces with constant curvature in Euclidean spaces.
problem Rigidity of hypersurfaces with constant mean and scalar curvature.
method Characterizations and rigidity results under various conditions of Gaussian-Kronecker and r-th mean curvatures. result Rigidity theorems for hypersurfaces in dimensions 4, 5, and 6, and general dimensions under pinching conditions.
The article examines entropy-information inequalities for continuous-time Markov chains under curvature-dimension conditions.
problem Proving Li-Yau inequalities and modified logarithmic Sobolev inequalities for reversible Markov chains.
method Introducing the CDΥ(κ,F) condition and deriving entropy-information inequalities. result Derives functional inequalities relating entropy to Fisher information.
Study stability of curvature-dimension condition for negative dimensions.
problem Stability of curvature-dimension condition with negative dimension parameters.
method Introduced CD(K, N)-condition for N < 0, defined distance d_{\mathsf{iKRW}}, proved convergence stability.
result Limit structure of converging metric measure spaces remains CD(K, N) for N < 0.
New sub-Riemannian spaces with boundary meet curvature-dimension condition.
problem Finding sub-Riemannian manifolds with boundary satisfying curvature-dimension condition.
method Constructing specific sub-Riemannian structures on half-spaces and hemispheres.
result Provided new examples of sub-Riemannian manifolds with boundary that meet RCD(K,N) condition. Study Lp boundedness of Riesz transform on differential forms for certain manifolds.
problem Investigate Lp-boundedness of the covariant Riesz transform on differential forms. method Analyze Lp-boundedness on weighted Riemannian manifolds under curvature-dimension and lower bound conditions. result Derive Calderón-Zygmund inequality for 1<p≤2 under curvature-dimension condition. The study proves a neighborhood theorem for mean curvature flow in higher dimensions.
problem Proving a canonical neighborhood theorem for mean curvature flow in higher dimensions.
method Proved a canonical neighborhood theorem for mean curvature flow of compact submanifolds in RN with a pinching condition. result Proved a canonical neighborhood theorem for mean curvature flow in dimensions n≥5. Study confirms equivalence in Heisenberg groups between curvature-dimension conditions and strong Brunn-Minkowski inequalities.
problem Equivalence between curvature-dimension conditions and strong Brunn-Minkowski inequalities in Heisenberg groups.
method Optimal transport and approximation techniques in sub-Riemannian Heisenberg group Hn, combined with previous works.
result Confirms the equivalence in Heisenberg groups between curvature-dimension conditions and strong Brunn-Minkowski inequalities.
The paper studies how submanifolds of a sphere evolve over time.
problem Evolution of pinched submanifolds in the sphere.
method High codimension mean curvature flow with pinching conditions.
result Convergence to a round point or totally geodesic sphere under pinching conditions.
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…
Researchers prove inequalities for reaction-diffusion systems using a new curvature-dimension condition.
problem Proving Li-Yau and Harnack inequalities for systems of linear reaction-diffusion equations.
method Introducing a hybrid curvature-dimension condition and proving a differential Harnack estimate.
result A Harnack inequality holds under the hybrid curvature-dimension condition CDhyb(0,d) with d<∞. The study establishes a curvature-dimension condition for discrete Markov chains.
problem Proving modified logarithmic Sobolev inequalities for discrete Markov chains.
method Identifying and proving a curvature-dimension inequality CDΥ(κ,∞), and showing its compatibility with diffusive settings. result The CDΥ condition preserves curvature bounds under tensorization and leads to Beckner inequalities. The paper characterizes curvature-dimension conditions and related inequalities on Riemannian manifolds.
problem Curvature-dimension conditions and related inequalities on Riemannian manifolds.
method Information-theoretic approach to study curvature-dimension condition, rigidity theorems, and entropy differential inequalities.
result Equivalence of curvature-dimension condition and entropy differential inequalities on Riemannian manifolds.
The curvature-dimension condition fails in sub-Finsler geometry, extending previous results in sub-Riemannian geometry.
problem The failure of the curvature-dimension condition in sub-Finsler geometry.
method Non-trivial adaptation of Juillet's work, introduction of new tools and ideas.
result The CD(K,N) condition does not hold in sub-Finsler geometry for various norms and measures. Almost-Riemannian manifolds fail to meet a synthetic curvature condition.
problem Proving almost-Riemannian manifolds do not satisfy the CD condition. method Developed a new strategy to contradict the 1-dimensional CD condition. result 2D and strongly regular almost-Riemannian manifolds do not satisfy CD(K,N) for any K and N. The paper examines stability of the Sobolev inequality in metric spaces with curvature dimension conditions.
problem Investigating stability of the Sobolev inequality in metric spaces with curvature dimension conditions.
method Assuming almost the same optimal constant, the paper shows that the cumulative distribution of almost extremal functions is close to that of an Aubin-Talenti bubble on the round sphere.
result Quantitative stability with sharp exponent for the Sobolev inequality in various curvature and dimension assumptions.
Geodesics found in spacetime satisfy curvature conditions.
problem Finding geodesics in spacetime satisfying specific curvature conditions.
method Proving existence of geodesics with entropic semiconvexity and uniform L∞ densities. result Existence of geodesics satisfying the timelike curvature-dimension condition.
Paper refines Einstein manifold result with cone curvature condition.
problem Closed Einstein manifolds with specific curvature conditions.
method Relaxing curvature condition to cone condition and proving manifold properties.
result Closed Einstein manifolds of dimension 4, 5, or ≥8 are either flat or round spheres under the cone curvature condition.
New findings show different cost functions yield equivalent curvature bounds.
problem Establishing equivalence of curvature bounds under various transport costs.
method Needle decomposition and localization technique for optimal transport.
result All CDp(K,N) conditions are equivalent for p>1. 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.…
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…
On Riemannian manifolds of dimension 4, for prescribed scalar curvature equation, under lipschitzian condition on the prescribed curvature, we have an uniform estimate for the solutions of the equation if we control their minimas.
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 …
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…
Conditions ensure constant curvature in negatively curved manifolds.
problem Ensuring constant curvature in negatively curved manifolds.
method Intrinsic conditions on horospheres' geometry.
result Sectional curvature is constant under given conditions.
Proposes a new metric space example showing non-constant topological dimension.
problem Non-constant topological dimension in metric measure spaces.
method Refines Ketterer and Rajala's example to satisfy CD(0,∞) condition.
result Shows non-constancy of topological dimension for CD spaces.
We show that a minimal disk satisfying the free boundary condition in a constant curvature ball of any dimension is totally geodesic. We weaken the condition to parallel mean curvature vector in which case we show that the disk lies in a three dimensional constant curvature submanifold and is totally umbilic. These res…
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…
Timelike curvature and Brunn-Minkowski inequality linked in non-smooth spacetimes.
problem Equivalence between timelike Ricci curvature and Brunn-Minkowski inequality in synthetic Lorentzian spaces.
method Introducing strong q-timelike Brunn-Minkowski condition and proving equivalence to curvature conditions. result Timelike curvature dimension condition equivalent to timelike Brunn-Minkowski inequality in specific settings.
We prove that codimension two surfaces satisfying a nonlinear curvature condition depending on normal curvature are smoothly deformed by mean curvature flow to round points.
Investigates maps and properties in spaces with negative dimensions and curvature.
problem Existence of transport maps and local-to-global property in spaces with negative dimensions and bounded Ricci curvature.
method Examines metric measure spaces with negative curvature dimensions and applies reduced curvature-dimension conditions.
result Establishes the existence of transport maps and proves the local-to-global property.
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…
In this note we characterize compact hypersurfaces of dimension n≥2 with constant mean curvature H immersed in space forms of constant curvature and satisfying an optimal integral pinching condition: they are either totally umbilical or, when n≥3 and H=0, they are locally contained in a rotational h…
Let (M,g_0) be a compact Riemannian manifold of dimension n \geq 4. We show that the normalized Ricci flow deforms g_0 to a constant curvature metric provided that (M,g_0) x R has positive isotropic curvature. This condition is stronger than 2-positive flag curvature but weaker than 2-positive curvature operator.
This note is a study of nonnegativity conditions on curvature which are preserved by the Ricci flow. We focus on specific kinds of curvature conditions which we call noncoercive, these are the conditions for which nonnegative curvature and vanishing scalar curvature doesn't imply flatness. We show that, in dimensions g…
We prove a gradient estimate for graphical spacelike mean curvature flow with a general Neumann boundary condition in dimension n=2. This then implies that the mean curvature flow exists for all time and converges to a translating solution.
Study curvature operators in 4n-dimensional manifolds, finding new conformal invariants.
problem Analyzing curvature operators in oriented Riemannian 4n-manifolds.
method Examining finite systems of hafnian identities in eigenvalues, focusing on locally conformally flat cases.
result Discovering a new conformal invariant in dimensions 4n, related to nonnegativity of Euler characteristic.
We study a positivity condition for the curvature of oriented Riemannian 4-manifolds: The half-PIC condition. It is a slight weakening of the positive isotropic curvature (PIC) condition introduced by M. Micallef and J. Moore. We observe that the half-PIC condition is preserved by the Ricci flow and satisfies a m…
Example of spacetime with causal bubbling, splitting into timelike and spacelike parts.
problem Understanding causal bubbling in spacetimes.
method Example of a globally hyperbolic spacetime with a continuous metric, splitting orthogonally into timelike and spacelike parts.
result The synthetic timelike curvature-dimension (TCD) condition does not prevent causal bubbling.