This survey introduces synthetic timelike Ricci curvature bounds in Lorentzian spaces.
problem Synthetic timelike Ricci curvature bounds in non-smooth Lorentzian spaces.
method Optimal transport and entropy tools.
result Synthetic version of Hawking's singularity theorem and synthetic characterisation of Einstein's vacuum equations.
Equivalence found between smooth and synthetic timelike curvature bounds.
problem Understanding timelike sectional curvature bounds in spacetime geometry.
method Established equivalence between sectional curvature bounds on timelike planes and synthetic timelike bounds.
result Equivalence proved for sectional curvature bounds on timelike planes and synthetic timelike bounds on strongly causal spacetimes.
Unified study of Riemannian and sub-Riemannian geometries with synthetic Ricci curvature bounds.
problem Unified framework for Riemannian and sub-Riemannian geometries.
method Study of gauge metric measure spaces.
result Unified synthetic Ricci curvature lower bounds for both Riemannian and sub-Riemannian structures.
Proves equivalence of two types of Ricci curvature bounds.
problem Equivalence of distributional and synthetic Ricci curvature bounds.
method Analyzes weighted Riemannian manifolds with specific smoothness conditions.
result Proves equivalence of Ricci curvature bounds under given conditions.
We study the fundamental properties of curvature in groupoids within the framework of synthetic differential geometry. As is usual in synthetic differential geometry, its combinatorial nature is emphasized. In particular, the classical Bianchi identity is deduced from its combinatorial one.
Study compares synthetic and distributional Ricci curvature bounds.
problem Comparing synthetic and distributional approaches to lower Ricci curvature bounds.
method Analyzes synthetic via weak displacement convexity and distributional via non-negativity of Ricci-tensor.
result Distributional bounds imply entropy bounds for C1 metrics and vice versa for C1,1 under convergence condition. New relation between curvature bounds and spacetime inextendibility.
problem Inextendibility of spacetimes under low regularity conditions.
method Synthetic curvature and causal character analysis.
result Low-regularity spacetimes with unbounded curvature.
New relation between curvature bounds and spacetime inextendibility.
problem Inextendibility of spacetimes under low regularity conditions.
method Synthetic curvature and causal character maximizers.
result Low-regularity inextendibility linked to unbounded curvature.
Entropy study on synthetic spaces with curvature bounds.
problem Entropy functional on synthetic spaces with curvature bounds.
method Rigorous justification of entropy formula, monotonicity, and rigidity properties; heat kernel bounds.
result Bounds for heat equation solutions on synthetic spaces.
The study proposes conjectures on limit spaces of Riemannian manifolds with Ricci curvature.
problem Understanding the regularity of limit spaces of Riemannian manifolds with Ricci curvature.
method Synthetic treatment of lower bounds on Ricci curvature for metric measure spaces.
result Several conjectures on the regularity of limit spaces.
Survey on curvature bounds and isoperimetric inequalities.
problem Interplay between curvature bounds and isoperimetric problems.
method Survey of classical and recent results.
result Recent developments on curvature bounds and isoperimetric problems.
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.
Alexandrov spaces with non-negative curvature are characterized by the matrix displacement convexity of an entropy tensor.
problem Characterizing non-negative curvature in Alexandrov spaces
method Constructing a parallel trivialization of the entropy tensor
result The entropy tensor is matrix displacement convex on Alexandrov spaces
Synthetic proof shows globally hyperbolic Lorentzian spaces with specific curvature are warped products.
problem Synthetic proof of rigidity for globally hyperbolic Lorentzian spaces.
method Synthetic geometry and warped product analysis.
result Spaces with specific curvature and distance realizer are warped products.
Develops optimal transport in Lorentzian spaces with synthetic curvature bounds.
problem Synthetic curvature bounds for Lorentzian spaces.
method Optimal transport, convexity analysis of entropy functionals.
result Synthetic notion of timelike Ricci curvature lower bounds.
Uniformly Euclidean metrics with isolated singularities on certain manifolds are Ricci flat and have nonnegative synthetic Ricci curvature.
problem Proving the existence of Ricci flat metrics with isolated singularities on specific manifolds.
method Demonstrating nonnegative synthetic Ricci curvature using the RCD(0, n) condition.
result Uniformly Euclidean metrics with isolated singularities on Mn=Tn#M0 are Ricci flat and extend smoothly over the singularity. Synthetic scalar curvature defined via Gaussian integrals, applied to manifolds and flows.
problem Defining scalar curvature for non-smooth spaces and flows.
method Gaussian integral approach to scalar curvature, applied to manifolds and flows.
result Characterizes Ricci flows as minimal super-Ricci flows.
We analyze Lorentzian spacetimes subject to curvature-dimension bounds using the Bakry-Émery-Ricci tensor. We extend the Hawking-Penrose type singularity theorem and the Lorentzian timelike splitting theorem to synthetic dimensions N≤1, including all negative synthetic dimensions. The rigidity of the timelike spli…
We prove that in two dimensions the synthetic notions of lower bounds on sectional and on Ricci curvature coincide.
Computed distortion coefficients for the α-Grushin plane.
problem Analyzing the distortion coefficients of the α-Grushin plane.
method Using generalised trigonometric functions and synthetic curvature conditions.
result Estimates for distortion coefficients and a curvature condition conjecture.
The Penrose theorem and Hawking's topology theorem are extended to weighted spacetimes.
problem Extending Penrose's singularity theorem and Hawking's topology theorem to weighted spacetimes.
method Using weighted null energy condition and synthetic dimension to generalize the theorems.
result Generalized versions of the Penrose and Hawking theorems hold under a weighted null energy condition.
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. 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.
Sharp inequality in spaces with non-negative Ricci curvature.
problem Proving a sharp isoperimetric inequality in metric measure spaces.
method Using volume entropy in non-compact metric measure spaces with non-negative synthetic Ricci curvature.
result Proved a sharp dimension-free isoperimetric inequality.
In this paper we investigate Lott-Sturm-Villani's synthetic lower Ricci curvature bound on Riemannian manifolds with boundary. We prove several measure rigidity results for some important functional and geometric inequalities, which completely characterize CD(K,∞) condition and non-collapsed ${\rm CD}(K, …
Stability of timelike Ricci bounds in low-regularity spacetimes.
problem Stability of synthetic timelike Ricci curvature bounds under C0-limits. method Constructing smooth approximations and analyzing limiting behavior via Lorentzian optimal transport.
result Impulsive gravitational waves satisfy synthetic timelike Ricci curvature lower bounds.
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.
Globalisation theorem for Lorentzian spaces with curvature bounds.
problem Synthetic geometric analysis of Lorentzian length spaces.
method Cat's cradle construction and synthetic geometry.
result An analogue of Toponogov's Globalisation Theorem for Lorentzian length spaces.
Survey on gluing constructions under lower curvature bounds.
problem Understanding lower curvature bounds in various geometric contexts.
method Analyzes gluing constructions in smooth and non-smooth settings.
result Provides conjectures and theorems on synthetic lower Ricci curvature bounds.
The study improves harmonic map theory for metric spaces with curvature bounds.
problem Harmonic maps between specific metric spaces with curvature constraints.
method Synthetic geometry, Optimal Transport, Heat Flow, viscosity theory.
result Established Lipschitz continuity and Bochner-Eells-Sampson inequality.
Study inradius collapsed manifolds with lower Ricci curvature bounds, proving properties of their limits.
problem Characterizing limits of inradius collapsed manifolds with lower Ricci curvature bounds.
method Analyzing families of manifolds with specific curvature and boundary conditions, proving properties of the limits.
result Limits of inradius collapsed manifolds have at most two boundary components and a lower Ricci curvature bound.
Study solves Gel'fand's inverse problem in non-smooth spaces with Ricci curvature bounds.
problem Determining a Riemannian manifold from heat kernel on subsets.
method Analyzes mRCD(K,N) spaces with synthetic Ricci curvature bounds. result Unique solvability of Gel'fand's inverse problem for compact mRCD(K,N) spaces. Study extends null distance concept to Lorentzian length spaces for spacetime analysis.
problem Understanding spacetime convergence and topology in Lorentzian geometry.
method Extend null distance concept to Lorentzian length spaces, study Gromov-Hausdorff convergence.
result First results on compatibility of null distance with synthetic curvature bounds in warped product Lorentzian length spaces.
Synthetic splitting theorem for Lorentzian spaces with non-negative curvature.
problem Proving a splitting theorem for globally hyperbolic Lorentzian length spaces with non-negative timelike curvature.
method Synthetic approach using triangle comparison and parallelity of timelike lines.
result Establishes a splitting of a neighborhood of a complete timelike line, leading to global inextendibility.
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 new Ricci bounds for metric measure spaces, preserving properties under time changes.
problem Extend Ricci bounds to non-synthetic spaces and understand their behavior under time changes.
method Introduce distribution-valued lower Ricci bounds BE1(κ,∞), prove equivalence with gradient estimates, and show preservation under time changes. result Distribution-valued Ricci bounds BE1(κ,∞) are preserved under arbitrary time changes and imply sharp gradient estimates. Refines d'Alembertian for signed Lorentz distance functions in metric measure spacetimes.
problem Exact representation and bounds of d'Alembertian for signed Lorentz distance functions.
method Metric geometry techniques, localization, Sobolev calculus.
result Distributional d'Alembertian is a signed measure with integration by parts formula.
Timelike curves in Lorentzian length spaces have a total curvature notion that agrees with smooth curves.
problem Estimating the length of timelike curves in Lorentzian length spaces.
method Introducing a synthetic timelike total curvature notion.
result Proving timelike curves of finite total curvature are rectifiable.
Study on cones over metric spaces with curvature bounds.
problem Establishing curvature bounds for cones over metric spaces.
method Developed a localization technique to prove synthetic curvature bounds.
result Riemannian and Lorentzian cones over CD-spaces satisfy MCP and vice versa.
New sub-Riemannian structures fail synthetic curvature bounds.
problem Failure of synthetic curvature bounds in sub-Riemannian geometry.
method New stability results for local MCP under quotients, applied to specific sub-Riemannian structures.
result Ideal sub-Riemannian structures can fail the MCP, generically for high dimensions and rank > 3.
We study the low-regularity (in-)extendibility of spacetimes within the synthetic-geometric framework of Lorentzian length spaces developed in [KS:17]. To this end, we introduce appropriate notions of geodesics and timelike geodesic completeness and prove a general inextendibility result. Our results shed new light on …
The paper studies moduli spaces of non-smooth metric structures with non-negative Ricci curvature.
problem Understanding moduli spaces of non-smooth metric structures with non-negative Ricci curvature.
method Relating convergence of RCD(0,N)-structures to their lifts, constructing Albanese and soul maps, proving their continuity, and constructing examples.
result Construction of moduli spaces with non-trivial rational homotopy groups.
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. In this paper, we introduce the weighted mixed (sectional, Ricci and scalar) curvature of a foliated (and almost-product) Riemannian manifold (M,g) equipped with a vector field X. We define several functions (qth Ricci type curvatures), which "interpolate" between the weighed sectional and Ricci curvatures. The n…
Motivated by Perelman's Pseudo Locality Theorem for the Ricci flow, we prove that if a Riemannian manifold has Ricci curvature bounded below in a metric ball which moreover has almost maximal volume, then in a smaller ball (in a quantified sense) it holds an almost-euclidean isoperimetric inequality. The result is actu…
Proves a synthetic Lorentzian Cartan-Hadamard theorem.
problem Formulates and proves a theorem for Lorentzian geometry.
method Uses an appropriate notion of local concavity for Lorentzian (pre-)length spaces.
result Establishes existence and uniqueness of timelike geodesics.
New topological restrictions found for spaces with nonnegative Ricci curvature.
problem Understanding topological properties of spaces with nonnegative Ricci curvature.
method Analyzing complete Riemannian manifolds and RCD(0,n) spaces, applying rigidity and vanishing theorems.
result Proved a Betti number rigidity theorem and a vanishing theorem for simplicial volume.
Measure contraction property is a synthetic Ricci curvature lower bound for metric measure spaces. We consider Sasakian manifolds with non-negative Tanaka-Webster Ricci curvature equipped with the metric measure space structure defined by the sub-Riemannian metric and the Popp measure. We show that these spaces satisfy…