Ricci limit spaces are semi-locally simply connected.
problem Understanding the topological properties of Ricci limit spaces.
method Demonstrating that for any loop in a specified radius, it can be contracted within a larger radius.
result Ricci limit spaces are semi-locally simply connected.
New examples show non-integer Hausdorff dimensions in collapsing spaces.
problem Understanding Hausdorff dimensions in collapsing Ricci limit spaces.
method Provided examples of spaces with irregular Hausdorff dimensions.
result Hausdorff dimension of singular set exceeds regular set's dimension.
New examples of Ricci limit spaces with mixed tangent cones.
problem Constructing Ricci limit spaces with mixed tangent cones.
method For any integers m≥n≥3, construct a Ricci limit space X_{m,n} with specific tangent cones.
result Found a new example of Ricci limit space with mixed tangent cones.
Study the topology of Ricci limit spaces using Gromov-Hausdorff limits.
problem Topology of Ricci limit spaces.
method Gromov-Hausdorff limits, slice theorem for isometric pseudo-group actions, uniform diameter bounds.
result Established semi-locally simply connected property and described universal cover.
The study extends convergence theorems for Ricci-limit spaces with bounded curvature.
problem Understanding convergence properties of Ricci-limit spaces with bounded curvature.
method Establishing C1,α-regularities and applying Fukaya's fibration theorem. result Optimal generalization of Fukaya's fibration theorem to C1,α limit spaces. Study on positive scalar curvature and its impact on Ricci limit spaces.
problem The influence of uniformly positive scalar curvature on Ricci limit spaces.
method Investigates uniformly positive scalar curvature on non-collapsed Ricci limit spaces.
result Proves a limit space splits at most n-2 lines or R-factors.
Optimal diameter estimates for 3D spaces with non-negative Ricci curvature.
problem Estimating the diameter of 3D spaces with non-negative Ricci curvature.
method Proving positive scalar curvature passes to Ricci limit spaces of non-negative curvature.
result Optimal Bonnet-Myers upper bound for 3D spaces.
We study the behavior under Gromov-Hausdorff convergence of the spectrum of weighted $\barpartial$-Laplacian on compact Kähler manifolds. This situation typically occurs for a sequence of Fano manifolds with anticanonical Kähler class. We apply it to show that, if an almost smooth Fano-Ricci limit space admits a Kähler…
We use Ricci flow to obtain a local bi-Holder correspondence between Ricci limit spaces in three dimensions and smooth manifolds. This is more than a complete resolution of the three-dimensional case of the conjecture of Anderson-Cheeger-Colding-Tian, describing how Ricci limit spaces in three dimensions must be homeom…
Study confirms conjectures on Ricci limit spaces and their topological properties.
problem Understanding the topological structure of noncollapsed Ricci limit spaces.
method Analysis of tangent cones and application of manifold recognition theorems.
result Cross-sections of tangent cones at points in 4D spaces are homeomorphic to a fixed spherical space form.
New examples of manifolds in tangent cones of non-collapsed Ricci limit spaces.
problem Uncertainty in homeomorphism types of tangent cones of non-collapsed Ricci limit spaces.
method Construction of limit spaces in all dimensions at least 5.
result Any finite collection of manifolds can appear as cross sections of tangent cones of the same point.
Let X be a non-collapsing Ricci limit space and let x∈X. We show that for any ε>0, there is r>0 such that every loop in Bt(x) is contractible in B(1+ε)t(x), where t∈(0,r]. In particular, X is semi-locally simply connected.
2-regular points found in spaces with lower Ricci curvature bound.
problem Characterizing points in spaces with lower Ricci curvature bound.
method Analyzing measured Gromov-Hausdorff limits of Riemannian manifolds.
result 2-regular points in interior geodesics of limit spaces are 2-rectifiable.
We construct a global homeomorphism from any 3D Ricci limit space to a smooth manifold, that is locally bi-Holder. This extends the recent work of Miles Simon and the second author, and we build upon their techniques. A key step in our proof is the construction of local "pyramid Ricci flows", existing on uniform region…
I survey some of the developments in the theory of Ricci flow and its applications from the past decade. I focus mainly on the understanding of Ricci flows that are permitted to have unbounded curvature in the sense that the curvature can blow up as we wander off to spatial infinity and/or as we decrease time to some s…
New examples show strong Kato limits can be branching and not satisfy known conditions.
problem Exploring the boundaries of strong Kato limits and their properties.
method Constructing specific examples of non-collapsed strong Kato limits.
result Found examples of strong Kato limits that are branching and do not satisfy CD(K,∞) or MCP(K,N) conditions. We prove that `volume cone implies metric cone' in the setting of RCD spaces, thus generalising to this class of spaces a well known result of Cheeger-Colding valid in Ricci-limit spaces.
The paper analyzes graph Laplacians on manifolds with curvature bounds and applies to non-collapsed spaces.
problem Analyzing spectral properties of graph Laplacians on manifolds with curvature constraints.
method Quantitative bounds on eigenvalues and eigenfunctions of graph Laplacians constructed from random variables on manifolds with uniform lower Ricci curvature bounds.
result Spectral convergence of graph Laplacians on manifolds with curvature bounds and in non-collapsed 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.
In this paper we study elliptic PDEs on compact Gromov-Hausdorff limit spaces of Riemannian manifolds with lower Ricci curvature bounds. In particular we establish continuities of geometric quantities, which include solutions of Poisson's equations, eigenvalues of Schrodinger operators, generalized Yamabe constants and…
Paper reconciles different Ricci flow approaches and proves weak solutions.
problem Proving weak solutions for Ricci flows with singularities.
method Introducing a novel hitting estimate for Brownian motion, compensating for lack of lower heat kernel bounds.
result Every noncollapsed limit of Ricci flows and singular Ricci flows are weak solutions.
Perimeter minimizers in curved spaces have a singular set no more than 5 dimensions.
problem Understanding the structure of minimizers in spaces with bounded Ricci curvature.
method Analysis of non-collapsed Ricci limit spaces with two-sided curvature bounds.
result The Hausdorff dimension of the singular set is at most \(N-5\).
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…
Study of splitting maps in Type I Ricci flows for understanding singular set structure.
problem Understanding the structure of the singular set in non-collapsed Ricci limit spaces.
method Construction and investigation of almost splitting maps on Ricci flows that are almost self-similar.
result Sharp splitting maps remain splitting maps at smaller scales under certain conditions.
Researchers create metrics on spheres with Ricci curvature ≥1, limiting to Grushin hemisphere.
problem Constructing metrics with Ricci curvature ≥1 on spheres.
method Sequence of Riemannian metrics on Sm+n with Ric≥1. result Gromov-Hausdorff limit of the sequence is the Grushin hemisphere.
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.
The goal of the paper is to study the angle between two curves in the framework of metric (and metric measure) spaces. More precisely, we give a new notion of angle between two curves in a metric space. Such a notion has a natural interplay with optimal transportation and is particularly well suited for metric measure …
Research shows RCD* spaces are semi-locally simply connected.
problem Understanding the topological properties of RCD* spaces.
method Proving semi-locally simply connected property for any point and radius.
result RCD* spaces are semi-locally simply connected.
The study establishes conditions for orientability in spaces with lower Ricci curvature bounds.
problem Conditions for orientability in spaces with lower Ricci curvature bounds.
method Equivalent characterizations of orientability using Ricci limit and RCD spaces.
result Four-manifolds with Ricci curvature bounded below and volume non-collapsing are uniformly locally orientable.
The paper constructs manifolds with infinite holes from a given manifold.
problem Creating manifolds with infinite holes from a given manifold.
method Constructing a sequence of (n+2)-dimensional manifolds (Mi,gi) with mRicgi>λ that approximate the original manifold (X,h) and have an infinite number of connected components. result The constructed manifolds (Xε) have dense boundary with an infinite number of connected components and no open subset topologically a manifold. This note is devoted to the study of sets of finite perimeter over RCD(K,N) metric measure spaces. Its aim is to complete the picture about the generalization of De Giorgi's theorem within this framework. Starting from the results of [2] we obtain uniqueness of tangents and rectifiability for the reduced boundary of …
Study shows properties of Gromov-Hausdorff limit of frame bundles for non-collapsed manifolds.
problem Characterizing the Gromov-Hausdorff limit of orthonormal frame bundles of non-collapsed manifolds with bounded Ricci curvature.
method Analysis of the Gromov-Hausdorff limit space of orthonormal frame bundles equipped with an almost canonical metric.
result The singular set of the limit space has codimension ≥4 and the complement contains an open and dense C1,α-Riemannian manifold. The paper shows how heat flow approximates area functional on specific geometric spaces.
problem Approximating the area functional on $\RCD(K,\infty)$ spaces.
method Using heat flow and properties of $\RCD(K,\infty)$ spaces.
result The area functional coincides with its relaxation in $\RCD(K,\infty)$ spaces.
Let Y be a Gromov-Hausdorff limit of complete Riemannian n-manifolds with Ricci curvature bounded from below. A point in Y is called k-regular, if its tangent is unique and is isometric to an k-dimensional Euclidean space. By \cite{B5}, there is k>0 such that the set of all k-regular point Rk h…
3D Ricci flows have bounded diameter before Type I singularities.
problem Bounding the diameter of 3D Ricci flows before Type I singularities.
method Introduced a neck-region concept and proved packing measure Ahlfors regularity.
result Uniformly bounded diameter up to Type I singular time.
Defines a geometric dimension for Lorentzian spaces, distinguishing spacelike and null subspaces.
problem No specific problem stated; focuses on defining a new geometric dimension.
method Introduces a one-parameter family of volume measures and a doubling condition for causal diamonds.
result Defines a geometric dimension for synthetic spacetimes, distinguishing between spacelike and null subspaces.
Study on removing sets and uniqueness of diffusion operators on various spaces.
problem Determining the effect of removing small sets on the self-adjointness and uniqueness of diffusion operators.
method Analyzes symmetric diffusion operators on metric measure spaces, proving a truncation result for potentials.
result Characterizes the critical size of removed sets and their effect on operator properties.
New rigidity theorem for sharp spectral gap in nonnegatively curved spaces.
problem Rigidity of sharp spectral gap in nonnegatively curved spaces.
method Mixing Sobolev theory and singular 1D-localization.
result Rigidity of λ=diam2π2 in compact RCD(0,N) spaces. New Weyl's laws discovered for compact spaces with Ricci curvature bounds.
problem Understanding growth rates of eigenvalues in compact spaces with Ricci curvature constraints.
method Developed new properties of α-Grushin halfplanes and analyzed singular sets of null capacities. result Established Weyl's laws with power growth and logarithmic corrections for compact spaces.
Proof of Reifenberg theorem in metric spaces, expanding on Cheeger and Colding's work.
problem Proving the Reifenberg theorem in metric spaces using Gromov-Hausdorff distance.
method Detailed proof of Cheeger and Colding's result, expanding on their arguments.
result BiLipschitz version of the Reifenberg theorem in metric spaces.
Study shows different fundamental groups for manifolds with same limit.
problem Understanding fundamental groups of manifolds with non-negative Ricci curvature.
method Constructed sequences of manifolds with specific properties.
result Found manifolds with same limit but different fundamental groups.
Study minimal hypersurfaces in manifolds with bounded Ricci curvature.
problem Angle estimate of distance functions from minimal hypersurfaces.
method Colding's method and Cheeger-Colding theory.
result Prove Frankel property for metric cones.
New theorem on flat tori stability using harmonic maps and Ricci flow.
problem Stability of flat tori under Ricci and scalar curvature bounds.
method Harmonic map heat flow, Ricci flow, and RCD theories.
result Gromov-Hausdorff stability theorem for flat 3-tori.
The aim of this paper is to provide new stability results for sequences of metric measure spaces (Xi,di,mi) convergent in the measured Gromov-Hausdorff sense. By adopting the so-called extrinsic approach of embedding all metric spaces into a common one (X,d), we extend the results of Gigli-Mondino-Savaré by prov…
We prove that a compact RCD∗(0,N) (or equivalently RCD(0,N)) metric measure space, (X,d,m), with $\diam X \le d$ and its first (nonzero) eigenvalue of the Laplacian (in the sense of Ambrosio-Gigli-Savaré) , λ1=d2π2, has to be a circle or a line segment with diameter, π. This compl…
Study of tangent spaces in diffeological spaces under Lie group actions.
problem Understanding tangent spaces in generalized spaces.
method Generalized tangent space construction and isomorphism proof.
result Internal tangent space isomorphic to stratified tangent space.
(1,1) non-L-space knots are foliar in 3D space.
problem Proving (1,1) non-L-space knots are foliar.
method Analyzing (1,1) non-L-space knots in S3 and lens spaces. result (1,1) non-L-space knots are persistently foliar.
Universal spaces for finite topological spaces simplify shape descriptions.
problem Describing shape properties of compact metric spaces.
method Inverse limits of finite spaces and Alexandroff extensions.
result Universal spaces simplify shape descriptions of compact metric spaces.