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.
The study restricts ancient, type-I, non-collapsing 2D mean curvature flows to spheres or cylinders.
problem Understanding blow-up limits of ancient solutions in mean curvature flow.
method Argument of Giga and Kohn to restrict flows to spheres or cylinders.
result Ancient, type-I, non-collapsing 2D mean curvature flows are restricted to spheres or cylinders.
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. Proves an ε-regularity theorem for Ricci flows, leading to new singularity estimates.
problem Analyzing singularity models in Fano Kähler-Ricci flows.
method Proves ε-regularity theorem and uses it to derive new estimates.
result Establishes new estimates for singularity models of Fano Kähler-Ricci flows.
Study finite singularities in G2 structure flows using Shi-type estimates.
problem Finite time singularities in G2 structure flows.
method Extend Shi-type estimates to G2 structure flows and prove κ-non-collapsing theorem.
result Prove finite time singularities of G2 structure flows.
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.
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.
We first proved a compactness theorem of the Kähler metrics, which confirms a prediction of Chen. Then we prove several eigenvalue estimates along the Calabi flow. Combining the compactness theorem and these eigenvalue estimates, we generalize the method developed by Chen-Li-Wang to prove the small energy theorems of t…
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.
The paper proves convergence of 4-manifolds with almost vanishing curvature.
problem Proving convergence of Riemannian 4-manifolds with vanishing curvature.
method Used L2-curvature flow and smoothing techniques. result Proves convergence to flat or Einstein manifolds.
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.
The paper extends Nakamaye's theorem to non-closed forms on complex manifolds.
problem Analyzing non-closed (1,1)-forms on compact complex manifolds. method Developed analytic technique by Collins and Tosatti to study non-Hermitian loci.
result Non-Hermitian locus equals union of positive-dimensional null subvarieties.
Study quantizes topological numbers on degenerating Einstein manifolds.
problem Quantizing topological numbers on non-collapsed degenerating Einstein manifolds.
method Compactness theory of bubbles, classical vanishing theorems, and Hirzebruch-Riemann-Roch theorems.
result Established quantization results for various topological numbers.
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.
New differential operator helps characterize non-collapsed RCD spaces.
problem Characterizing non-collapsed compact RCD spaces.
method Explicit formula of Laplacian associated with pull-back Riemannian metric via heat kernel.
result Confirms conjecture on implication from weakly non-collapsed to non-collapsed condition.
Characterizes limits of Ricci flows and their singularities.
problem Understanding the structure of non-collapsed limits of Ricci flows.
method Characterizes limits as smooth away from a set of high codimension, identifies tangent flows as gradient shrinking solitons, and stratifies singular set.
result Non-collapsed limits of Ricci flows are smooth away from a set of high codimension and have tangent flows as gradient shrinking solitons.
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.
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. 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.
The paper studies limits of sub-Riemannian Heisenberg manifolds and proves convergence under certain conditions.
problem The study of limits of sub-Riemannian Heisenberg manifolds and their properties.
method Analysis of Gromov-Hausdorff limits with upper and lower bounds on diameter and Popp's measure.
result The non-collapsed limits of sub-Riemannian Heisenberg manifolds converge to a compact Heisenberg manifold.
Non-collapsing Ricci limit spaces are shown to be semi-locally simply connected.
problem Understanding the topological properties of non-collapsing Ricci limit spaces.
method Demonstrated through the existence of a radius r for each point x in the space, such that loops are contractible within a larger radius. result Non-collapsing Ricci limit spaces are semi-locally simply connected.
Develops structure theory for Ricci shrinkers without curvature restrictions.
problem Understanding the structure of Ricci shrinkers without curvature conditions.
method Structure theory development for non-collapsed Ricci shrinkers.
result Curvature estimates of Ricci shrinkers based on non-collapsing constant.
Compactness theorem for Riemannian manifolds with volume and curvature bounds.
problem Investigating the regularity of limit spaces of Riemannian manifolds.
method Local volume growth condition, compactness theorem, different convergence notion.
result Compactness theorem for Riemannian manifolds with Lp curvature bounds and volume growth assumption. We prove an existence theorem for Asymptotically Conical Ricci Flat Kahler metrics in C2 with cone singularities along a smooth complex curve. These metrics are expected to arise as blow up limits of non collapsed sequences of Kahler Einstein metrics with cone singularities.
Two new proofs show Ricci flow breathers are special solutions.
problem Characterize solutions to Ricci flow on closed manifolds.
method Use singularity models and ancient solutions to show they are gradient Ricci solitons.
result Ricci flow breathers are gradient Ricci solitons.
We prove a non-collapsing property for curvature flows of embedded hypersurfaces in the sphere and in hyperbolic space.
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.
New steady Ricci solitons found in even dimensions, including a four-dimensional example.
problem Existence of non-collapsed steady Ricci solitons in even dimensions.
method Construction of a family of non-collapsed, non-Kähler, non-Einstein steady Ricci solitons on complex line bundles over Kähler-Einstein manifolds.
result Existence of new non-collapsed steady Ricci solitons in even dimensions, including a four-dimensional example.
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.
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.
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…
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.
The paper studies 4D Ricci flow manifolds with curvature constraints.
problem Investigating 4D Ricci flow manifolds with specific curvature conditions.
method Analyzing 4D manifolds with curvature constraints via Ricci flow.
result Proves topological and geometric gap theorems for maximal volume growth.
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 paper examines the rigidity of eigenvalues in shrinking Ricci solitons.
problem Rigidity of eigenvalues in shrinking Ricci solitons.
method Analysis of the drifted Laplacian on shrinking Ricci solitons, showing eigenvalue bounds and rigidity results.
result If the nextth eigenvalue is close to a lower bound, the n-soliton must be the trivial Gaussian soliton. 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.
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.
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. 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.
The paper proves compactness and structure of surfaces with small curvature.
problem Compactness and local structure of surfaces with small total curvature.
method Introduced a new quantity called isothermal radius to establish compactness in intrinsic and extrinsic topologies.
result Established a compactness theorem for surfaces in intrinsic Lp-topology and extrinsic W2,2-weak topology. 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.
The paper explores sharp isoperimetric properties on non-compact spaces with Ricci bounds.
problem Sharp isoperimetric properties on non-compact spaces with Ricci bounds.
method Sharp isoperimetric comparison theorems and asymptotic isoperimetric properties.
result Almost regularity theorems and enhanced functional inequalities.
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 …
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…