This paper shows moduli spaces of RCD(0,2) structures are contractible.
problem Understanding moduli spaces of RCD(0,2) structures.
method Established a list of compact topological spaces admitting RCD(0,2) structures and described their associated moduli spaces.
result All moduli spaces of RCD(0,2) structures are contractible.
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.
Study cohomogeneity one RCD-spaces, proving structural results and constructing new examples.
problem Characterize and construct RCD-spaces with cohomogeneity one actions.
method Slice Theorem, construction from group diagrams, topological structural results.
result Classification of cohomogeneity one, non-collapsed RCD-spaces of essential dimension at most 4.
Study on RCD(0,N) spaces with small linear diameter growth.
problem Understanding structure properties of RCD(0,N) spaces.
method Analyzing the (revised) fundamental group of RCD(0,N) spaces.
result Proved that the revised fundamental group is finitely generated for RCD(0,N) spaces with small linear diameter growth.
Compact RCD spaces derived from singular Kahler metrics on 3D projective varieties.
problem Understanding geometric structures of singular Kahler spaces.
method Proving RCD spaces homeomorphic to 3D projective varieties with bounded Nash entropy and Ricci curvature.
result Compact RCD spaces are equivalent to underlying projective varieties.
Paper develops a splitting principle for RCD spaces, extending manifold properties.
problem Understanding splitting properties in RCD spaces.
method General analytic splitting principle for RCD spaces.
result Spaces with suitable functions have splitting properties.
The main goal of the paper is to prove the existence of the universal cover for RCD∗(K,N)-spaces. This generalizes earlier work of C. Sormani and the second named author on the existence of universal covers for Ricci limit spaces. As a result, we also obtain several structure results on the (revised) fundamental gro…
Mondino and Naber recently proved that finite dimensional RCD spaces are rectifiable. Here we show that the push-forward of the reference measure under the charts built by them is absolutely continuous with respect to the Lebesgue measure. This result, read in conjunction with another recent work of us, has relev…
Minimal surface equation results in constant solutions on RCD spaces.
problem Analyzing minimal surfaces on RCD spaces.
method Using properties of RCD spaces and the minimal surface equation.
result Positive solutions to the minimal surface equation are constant on RCD spaces.
We develop a structure theory for RCD spaces with curvature bounded above in Alexandrov sense. In particular, we show that any such space is a topological manifold with boundary whose interior is equal to the set of regular points. Further the set of regular points is a smooth manifold and is geodesically convex. Aroun…
The paper describes a fine representation of the Ricci tensor and Hessian on RCD spaces.
problem Understanding the structure of the Ricci tensor and Hessians on RCD spaces.
method Polar decomposition of the Ricci tensor and Hessians, providing regularity results.
result The Ricci tensor and Hessians on RCD spaces can be represented by a polar decomposition, revealing their regularity.
For n-dimensional Riemannian manifolds M with Ricci curvature bounded below by −(n−1), the volume entropy is bounded above by n−1. If M is compact, it is known that the equality holds if and only if M is hyperbolic. We extend this result to RCD∗(−(N−1),N) spaces. While the upper bound is st…
New Einstein RCD spaces found with cone singularities.
problem Existence of Einstein RCD spaces with cone singularities.
method Characterization of RCD spaces and cone singularity analysis.
result Existence of smooth non-compact 4-manifolds with ALE Ricci-flat RCD(0,4) metrics.
Study fundamental groups of RCD spaces without smoothness or curvature bounds.
problem Understanding fundamental groups of RCD spaces without additional conditions.
method Combining tools from RCD spaces, Gromov-Hausdorff convergence, and splitting theorems.
result Fundamental groups of RCD spaces are controlled by a finite number of generators and have specific properties under convergence.
Paper proves weak unique continuation for harmonic functions on RCD spaces but finds counterexample for strong uniqueness.
problem Unique continuation of harmonic functions on RCD spaces, especially strong uniqueness.
method Establishes weak unique continuation theorem and provides counterexample for strong uniqueness.
result Found counterexample for strong unique continuation in RCD(K,N) spaces for N≥4 and K∈R.
We construct non-symmetric diffusion processes associated with Dirichlet forms consisting of uniformly elliptic forms and derivation operators with killing terms on RCD spaces by aid of non-smooth differential structures introduced by Gigli '16. After constructing diffusions, we investigate conservativeness and the wea…
Characterizes when almost smooth spaces become RCD spaces.
problem Understanding conditions for almost smooth spaces to be RCD spaces.
method Characterizations via local volume doubling and Poincaré inequality.
result Characterizes Einstein 4-orbifolds.
We consider a rigidity problem for the spectral gap of the Laplacian on an RCD(K,∞)-space (a metric measure space satisfying the Riemannian curvature-dimension condition) for positive K. For a weighted Riemannian manifold, Cheng--Zhou showed that the sharp spectral gap is achieved only when a 1-dimensional G…
Extends Margulis Lemma to RCD(K,N) spaces.
problem Applying Margulis Lemma to new geometric structures.
method Improved Regularity Estimates for Regular Langrangian Flows.
result Margulis Lemma extended to RCD(K,N) spaces.
Study spectral distances on compact RCD spaces.
problem Understanding spectral convergence in RCD spaces.
method Established relationships between different spectral convergences and constructed a spectral approximation map.
result Found canonical spectral approximation map for RCD spaces.
Causal discovery from data affected by latent confounders is an important and difficult challenge. Causal functional model-based approaches have not been used to present variables whose relationships are affected by latent confounders, while some constraint-based methods can present them. This paper proposes a causal f…
Extended Rank-One Theorem to special metric spaces.
problem Extending a theorem to new types of spaces.
method Applied to a new class of metric measure spaces.
result Rank-One Theorem proven for RCD(K,N) spaces. In this paper, we prove pointwise convergence of heat kernels for mGH-convergent sequences of RCD∗(K,N)-spaces. We obtain as a corollary results on the short-time behavior of the heat kernel in RCD∗(K,N)-spaces. We use then these results to initiate the study of Weyl's law in the RCD setting
Locally homogeneous RCD spaces are shown to be smooth manifolds.
problem Understanding the structure of RCD spaces.
method Adapting existing results to new spaces.
result Locally homogeneous RCD spaces are isometric to smooth manifolds.
Paper extends Weyl's lemma to RCD(K,N) spaces.
problem Applying Weyl's lemma to RCD(K,N) metric measure spaces.
method Extending Weyl's lemma to RCD(K,N) spaces and proving applications.
result Local regularity of solutions for Poisson equations and Liouville-type results for harmonic functions.
This study proves the Half Space Property for RCD(0,N) and RCD(K,N) spaces.
problem Proving the Half Space Property for RCD(K,N) spaces.
method Analyzing locally perimeter minimizing sets and extending Green's functions results.
result The Half Space Property holds for RCD(K,N) spaces under specific conditions.
Researchers extend monotonicity formulas for harmonic functions in RCD(0,N) spaces.
problem Generalizing monotonicity formulas for harmonic functions in mRCD(0,N) spaces. method New estimates for harmonic functions and a functional version of the outer volume cone theorem.
result Proven rigidity and almost rigidity statements for harmonic functions in mRCD(0,N) spaces. New isoperimetric inequality for clamped plates in RCD(0,N) spaces, sharp and stable.
problem Fine properties of the principal frequency of clamped plates in RCD(0,N) spaces.
method Analyzing the RCD(0,N) spaces and applying isoperimetric inequalities.
result Sharp isoperimetric inequality for the principal frequency of clamped plates in RCD(0,N) spaces.
Since non-compact RCD(0, N) spaces have at least linear volume growth, we study noncompact RCD(0, N) spaces with linear volume growth in this paper. One of the main results is that the diameter of level sets of a Busemann function grow at most linearly on a noncompact RCD(0, N) space satisfying the linear volume growth…
We prove topological sphere theorems for RCD(n-1, n) spaces which generalize Colding's results and Petersen's result to the RCD setting. We also get an improved sphere theorem in the case of Einstein stratified spaces.
The paper shows equivalent interpretations of Laplacian bounds in RCD spaces.
problem Equivalence of Laplacian bounds in RCD spaces.
method Analyzes different interpretations of the inequality Δf ≤ η in RCD(K,N) spaces.
result Improves generality and regularity assumptions for function f.
The paper proves properties of non-collapsed RCD spaces with bounded covering geometry.
problem Characterizing properties of non-collapsed RCD spaces.
method Analyzing local covering geometry and applying Gromov's almost flat manifold theorem.
result RCD spaces with bounded covering geometry are biHölder homeomorphic to infranil-manifolds.
Study examines harmonic functions in sub-Riemannian and RCD settings.
problem Characterizing harmonic functions in sub-Riemannian and RCD settings.
method Analyzes weak and strong asymptotically mean value harmonic functions.
result Weakly amv-harmonic functions are equivalent to harmonicity in Carnot groups.
Compact RCD spaces are proven to be smooth manifolds under specific harmonicity conditions.
problem Characterizing compact RCD spaces as harmonic manifolds.
method Analyzing heat kernel and geodesic ball volumes for harmonicity.
result Compact RCD spaces are isometric to smooth manifolds under given conditions.
Researchers find second-order estimates for p-Laplacian in RCD spaces.
problem Estimating functions with p-Laplacian in RCD spaces. method Establishing quantitative second-order Sobolev regularity.
result Second-order estimates for p-Laplacian functions in RCD spaces. We show that on every RCD spaces it is possible to introduce, by a distributional-like approach, a Riemann curvature tensor. Since after the works of Petrunin and Zhang-Zhu we know that finite dimensional Alexandrov spaces are RCD spaces, our construction applies in particular to the Alexandrov setting.…
Uniform bounds prove connection between Kähler metrics and RCD spaces.
problem Bounding Nash entropy and Calabi energy for Kähler metrics.
method Proving uniform Sobolev bounds for Kähler manifolds.
result Establishes connection to RCD spaces and provides examples.
Proves isometric embeddings in Euclidean spaces for RCD spaces.
problem Isometric immersions of RCD spaces in Euclidean spaces.
method Analyzes regular isometric immersions and eigenmaps of compact non-collapsed RCD spaces.
result Eigenmaps of compact non-collapsed RCD spaces are locally bi-Lipschitz embeddings to spheres.
Study shows how maps from certain geometric spaces behave near their edges.
problem Boundary regularity of harmonic maps in specific geometric spaces.
method Analysis of RCD(K,N) and CAT(0) spaces. result Established boundary regularity of harmonic maps.
The paper develops techniques to study entropy and rigidity in RCD-spaces.
problem Entropy and rigidity in RCD-spaces.
method Develops the barycenter technique for RCD-spaces and applies it to show entropy-volume inequalities.
result RCD-spaces with equality in entropy-volume inequality are locally symmetric.
Sharp inequalities proved for RCD spaces, showing equality conditions.
problem Proving sharp inequalities for RCD spaces and identifying equality conditions.
method Analyzing RCD(1,∞) and RCD(K,∞) spaces to prove inequalities and identify equality conditions. result Equality conditions for Buser's and Cheeger's inequalities in RCD spaces.
Study on heat equation and eigenfunctions on RCD spaces, proving unique continuation.
problem Unique continuation for caloric functions and eigenfunctions on RCD spaces.
method Establish weak unique continuation theorem for caloric functions and eigenfunctions on compact RCD(K,2) spaces.
result Existence of non-trivial eigenfunctions and caloric solutions vanishing up to infinite order at one point.
Proves weakly non-collapsed RCD spaces are strongly non-collapsed.
problem Proving the equivalence of weakly non-collapsed and strongly non-collapsed RCD spaces.
method Analyzes properties of RCD spaces and uses auxiliary results.
result Confirms conjecture about RCD spaces being strongly non-collapsed.
We establish topological regularity and stability of N-dimensional RCD(K,N) spaces (up to a small singular set), also called non-collapsed RCD(K,N) in the literature. We also introduce the notion of a boundary of such spaces and study its properties, including its behavior under Gromov-Hausdorff convergence.
Study heat content on RCD(K,N) spaces with specific boundary conditions.
problem Analyzing heat content in RCD(K,N) spaces with irregular boundaries.
method Proved first-order asymptotics using measured interior geodesic condition.
result Established first-order heat content asymptotics on RCD(K,N) spaces.
Study Kähler-Einstein metrics on singular varieties, proving metric completion properties.
problem Kähler-Einstein metrics on singular projective varieties.
method Approximation with constant scalar curvature metrics, RCD space analysis.
result Metric completion of smooth part is non-collapsed RCD space and homeomorphic to original variety.
Paper finds formulas for minimizing perimeter in special spaces, proving key dimensions and existence.
problem Understanding the structure of perimeter minimizing sets in specific metric spaces.
method Established a monotonicity formula and proved rigidity for perimeter minimizers in RCD(0,N) spaces.
result Sharp Hausdorff dimension estimates for singular strata and existence of blow-down cones.
Heat kernels map RCD spaces to Riemannian manifolds.
problem Mapping RCD spaces to Riemannian manifolds using heat kernels.
method Using heat kernels to map RCD spaces into L2 space and then normalizing to achieve isometric immersions. result Compact RCD spaces with isometrically heat kernel immersions are isometric to unweighted smooth Riemannian manifolds.