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.
Topology of non-orientable spaces without boundary is studied.
problem Topology of non-collapsed RCD spaces without boundary.
method Studied the stability of non-orientability and topology under Gromov-Hausdorff convergence.
result Non-orientable spaces without boundary have a stable ramified double cover.
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…
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. We prove a non-collapsing property for curvature flows of embedded hypersurfaces in the sphere and in hyperbolic space.
Study characterizes non-collapsed RCD(K, N) spaces using heat kernel metrics.
problem Characterize non-collapsed RCD(K, N) spaces via heat kernel metrics.
method Investigate the second principal term in heat kernel metrics and prove divergence free property.
result Proves non-collapsed property via divergence free property of heat kernel metrics.
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.
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.
Theory of parallel transport on non-collapsed RCD spaces established.
problem Parallel transport on non-collapsed RCD spaces.
method General theory developed for parallel transport on non-collapsed RCD spaces, including geodesics and curves via time-dependent vector fields.
result Existence and uniqueness of parallel transport results obtained.
Sphere theorems for specific manifolds with curvature constraints.
problem Sphere theorems for Riemannian manifolds with scalar curvature bounds and non-collapsed RCD(n−1,n) spaces. method Analysis of scalar curvature and mean distance constraints.
result Established sphere theorems for the specified manifolds.
Study Busemann spaces with measures under MCP, proving rigidity and structure theorems.
problem Understanding the structure of Busemann spaces with measures.
method Analyzing geodesic completeness and non-collapse assumptions.
result Rigidity and structure theorems for Busemann spaces with MCP.
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. Study shows local topologies of certain geometric spaces.
problem Local topological properties of geometric spaces.
method Analysis of Gromov-Hausdorff limits of manifolds with bounded Ricci curvature.
result Local b1 vanishes for regular loci in limits of non-collapsed manifolds. 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.
Based on the compactness of the moduli of non-collapsed Calabi-Yau spaces with mild singularities, we set up a structure theory for polarized Kähler Ricci flows with proper geometric bounds. Our theory is a generalization of the structure theory of non-collapsed Kähler Einstein manifolds. As applications, we prove the …
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 \(N-5\).
Formula connects curvature to volume in special geometric spaces.
problem Deriving formulas for curvature in specific geometric spaces.
method Used strong locality of Laplacian and eigenfunction approximation.
result Proved integral type Gauss-Green formula linking curvature to volume.
Solves capillary Lp-Christoffel-Minkowski problem in half-space.
problem Capillary surfaces in half-space geometry.
method Non-collapsing estimate for height and capillary support function.
result Extends Christoffel-Minkowski existence result.
Study shows dimension constraints for isometry groups in non-collapsed Riemannian manifolds.
problem Dimension constraints for isometry groups in non-collapsed Riemannian manifolds.
method Equivariant Gromov--Hausdorff convergence and lower Ricci curvature bounds.
result Dimension of isometry group is at least the limit superior of dimensions of subgroups.
Uniform entropy bound for Ricci shrinkers with bounded curvature.
problem Bounding entropy for Ricci shrinkers with specific curvature constraints.
method Establishing uniform entropy bounds for simply connected Ricci shrinkers with a finite second homotopy group and uniform curvature bounds.
result Uniform entropy bound for simply connected Ricci shrinkers with a finite second homotopy group and uniform curvature bounds.
It was recently proved that embedded solutions of Euclidean hypersurface flows with speeds given by concave (convex), degree one homogeneous functions of the Weingarten map are interior (exterior) non-collapsing. These results were subsequently extended to hypersurface flows in the sphere and hyperbolic space. In the f…
Uniform volume estimate for Kähler metrics in big cohomology classes.
problem Estimating volume for singular Kähler metrics in big cohomology classes.
method Generalized mixed energy estimate for functions in complex Sobolev space to big cohomology classes.
result Uniform non-collapsing volume estimate for local Kähler metrics.
Ancient Ricci flows on non-collapsed manifolds have finite fundamental groups.
problem Understanding the fundamental groups of ancient Ricci flows.
method Analyzing the structure of ancient Ricci flows and their tangent flows.
result The fundamental group of non-collapsed ancient Ricci flows is finite and a quotient of the regular part's fundamental group.
Researchers describe a spectral sequence for knots in 3D space.
problem Understanding the Sinha spectral sequence for knots in R^3.
method Explicit description using Fox Neuwirth chain complexes and multicomplex structure.
result A non-trivial third page differential found, contradicting the rational case.
Study on metric spaces with properties (ETR), (LBD) and their convergence.
problem Understanding orientability and convergence of metric measure spaces.
method Analysis of Gromov-Hausdorff and intrinsic flat convergence for spaces satisfying (ETR), (LBD).
result The pointed Gromov-Hausdorff limit coincides with the local flat limit.
The paper applies a capillary John ellipsoid theorem to solve capillary curvature problems.
problem Solving capillary curvature problems in Euclidean half-spaces.
method Applying a capillary John ellipsoid theorem to derive non-collapsing estimates and gradient estimates.
result Established existence of solutions to capillary curvature problems in certain ranges of p and q. 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 two types of singular Kähler-Einstein metrics on complex varieties.
problem Understanding different types of singular Kähler-Einstein metrics.
method Analyzing Ricci flat Kähler cone metrics and applying to more general spaces.
result A weaker notion of singular Kähler-Einstein metrics is equivalent to a stronger one under certain conditions.
We prove a uniform Sobolev inequality for Ricci flow, which is independent of the number of surgeries. As an application, under less assumptions, a non-collapsing result stronger than Perelman's κ non-collapsing with surgery is derived. The proof is shorter and seems more accessible. The result also improves some ear…
We show non-collapsing for the evolution of nearly spherical closed convex curves in \mathbb{R}^2 under power curvature flow using two-point-methods.
The study shows almost maximal volume entropy rigidity for certain manifolds with integral Ricci curvature.
problem Volume entropy rigidity for manifolds with lower integral Ricci curvature bound.
method Analyzing manifolds with specific integral Ricci curvature bounds, diameter, and volume entropy.
result The universal cover of the manifold is close to a hyperbolic space form under certain conditions.
We develop a structure theory for non-collapsed Ricci shrinkers without any curvature condition. As applications, we obtain some curvature estimates of the Ricci shrinkers depending only on the non-collapsing constant.
Researchers found infinitely many non-collapsed steady Ricci solitons on complex line bundles.
problem Finding steady Ricci solitons on complex line bundles.
method Constructed a continuous 3-parameter family of non-shrinking Ricci solitons.
result Includes infinitely many asymptotically paraboloidal steady Ricci solitons.
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.
In this short note we provide several conjectures on the regularity of measured Gromov-Hausdorff limit spaces of Riemannian manifolds with Ricci curvature bounded below, from the point of view of the synthetic treatment of lower bounds on Ricci curvature for metric measure spaces.
Extends Ricci flow theory to Kato-type curvature bounds, proving manifold properties.
problem Extending Ricci flow theory under Kato-type curvature bounds.
method Extends non-collapsed Ricci flow existence theory to Kato-type lower bounds.
result Compact three-dimensional non-collapsed strong Kato limit spaces are homeomorphic to smooth manifolds.
Ricci flow singularities on compact Kähler surfaces are of Type I.
problem Understanding finite time singularities of Ricci flow on compact Kähler surfaces.
method Analyzing the Type I property of singularities.
result Non-collapsed finite time singularities are of Type I.
Paper proves Allard's theorem in Alexandrov spaces.
problem Proving Allard's theorem in non-collapsed Alexandrov spaces.
method Developed an intrinsic proof for Riemannian manifolds, then extended to Alexandrov spaces using approximation theorem.
result Explicit constants for constants in terms of geometric data.
We construct a family of non-collapsed, non-Kähler, non-Einstein steady Ricci solitons in even dimensions greater or equal to four. These solitons exist on complex line bundles over Kähler-Einstein manifolds of positive scalar curvature. They include a four-dimensional U(2)-invariant, non-collapsed Riemannian steady …
Study describes limits of non-collapsing K3 surfaces using algebraic data.
problem Understanding limits of non-collapsing polarized K3 surfaces.
method Explicit description via period mapping and algebro-geometric data.
result Bubbling limits depend solely on algebro-geometric data.
We prove a Lipschitz-Volume rigidity theorem for the non-collapsed Gromov-Hausdorff limits of manifolds with Ricci curvature bounded from below. This is a counterpart of the Lipschitz-Volume rigidity in Alexandrov geometry.
We consider complete (possibly non-compact) three dimensional Riemannian manifolds (M,g) such that: a) (M,g) is non-collapsed, b) the Ricci curvature of (M,g) is bounded from below, c) the geometry of (M,g) at infinity is not too extreme. Given such initial data (M,g) we show that a Ricci flow exists for a short time i…
We construct and classify, in the case of two complex dimensions, the possible tangent cones at points of limit spaces of non-collapsed sequences of Kähler-Einstein metrics with cone singularities.
Uniform estimates for Kaehler metrics' diameters and volumes.
problem Estimating diameters and volumes of Kaehler metrics.
method Proving uniform diameter and volume estimates for a family of Kaehler metrics.
result Uniform estimates for diameters and volumes of Kaehler metrics.
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.
We study non-collapsed Gromov-Hausdorff limits of Kähler manifolds with Ricci curvature bounded below. Our main result is that each tangent cone is homeomorphic to a normal affine variety. This extends a result of Donaldson-Sun, who considered non-collapsed limits of polarized Kähler manifolds with two-sided Ricci curv…
We provide a direct proof of a non-collapsing estimate for compact hypersurfaces with positive mean curvature moving under the mean curvature flow: Precisely, if every point on the initial hypersurface admits an interior sphere with radius inversely proportional to the mean curvature at that point, then this remains tr…
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.