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.
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.
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.
Warped products over one-dimensional base spaces satisfy curvature-dimension condition under specific conditions.
problem Proving the Riemannian curvature-dimension condition for warped products.
method Analyzing the function f and conditions on the base space and fiber.
result The Riemannian curvature-dimension condition is valid under specific constraints.
In this note we give necessary and sufficient conditions for the validity of the local spectral convergence, in balls, on the RCD∗-setting.
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 show characterizations of non-collapsed compact RCD(K,N) spaces, which in particular confirm a conjecture of De Philippis-Gigli on the implication from the weakly non-collapsed condition to the non-collapsed one in the compact case. The key idea is to give the explicit formula of the Laplacian associated to the p…
The study connects Kato bounds to finite-dimensional RCD spaces.
problem Understanding the limits of complete Riemannian manifolds with Kato bounds.
method Using the transformation rule of the Bakry-Émery condition under time change.
result Bi-Lipschitz equivalence to finite-dimensional RCD spaces.
We prove that the group of isometries of a metric measure space that satisfies the Riemannian curvature condition, RCD∗(K,N), is in fact a Lie group. We obtain an optimal upper bound on its dimension and classify the spaces where this maximal dimension is achieved.
Study on free boundary problems in RCD spaces, proving existence and regularity.
problem Free boundary problems in RCD metric measure spaces.
method Existence and local Lipschitz regularity of solutions, free boundary analysis.
result Existence and regularity of solutions, free boundary structure.
Sharp Talenti-type comparison theorem for p-Laplacian on RCD(K,N) spaces.
problem Understanding the p-Laplacian on RCD(K,N) spaces.
method Proving a Talenti-type comparison theorem.
result Sharp, rigid and stable Talenti-type comparison theorem.
Given a metric measure space (X,d,m) that satisfies the Riemannian Curvature Dimension condition, RCD∗(K,N), and a compact subgroup of isometries G≤Iso(X) we prove that there exists a G−invariant measure, mG, equivalent to m such that (X,d,mG) is still a…
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…
In this paper, we will study the (linear) geometric analysis on metric measure spaces. We will establish a local Li-Yau's estimate for weak solutions of the heat equation and prove a sharp Yau's gradient gradient for harmonic functions on metric measure spaces, under the Riemannian curvature-dimension condition $RCD^*(…
New estimates show spectral gap stability in RCD spaces, close to Beta distribution.
problem Stability of spectral gap bounds in metric-measure spaces.
method Combines L1-functional inequality and Stein's method. result Sharp quantitative estimate for spectral gap stability.
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.
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 Hölder continuity of tangent cones in RCD(K,N) spaces.
problem Understanding the geometry of metric measure spaces with curvature-dimension condition.
method Developed a second order interpolation formula for distance function.
result Tangent cones from rescalings are Hölder continuous along geodesics.
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.
We give necessary and sufficient conditions that show that both the group of isometries and the group of measure-preserving isometries are Lie groups for a large class of metric measure spaces. In addition we study, among other examples, whether spaces having a generalized lower Ricci curvature bound fulfill these requ…
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.
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.
In this short note, we give a sufficient condition for almost smooth compact metric measure spaces to satisfy the Bakry-Émery condition BE(K,N). The sufficient condition is satisfied for the glued space of any two (not necessary same dimensional) closed pointed Riemannian manifolds at their base points. This tells …
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.
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.
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.
The main result of this article states that the (K;N)-cone over some metric measure space satisfies the reduced Riemannian curvature-dimension condition RCD^*(KN;N+1) if and only if the underlying space satisfies RCD^*(N-1;N). The proof uses a characterization of reduced Riemannian curvature-dimension bounds by Bochner…
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.
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.
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.
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.
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. 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.
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.
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.
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.
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.
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.
We prove that a compact stratied space satises the Riemannian curvature-dimension condition RCD(K, N) if and only if its Ricci tensor is bounded below by K ∈ R on the regular set, the cone angle along the stratum of codimension two is smaller than or equal to 2π and its dimension is at most equal to N. This gives…