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 flat manifolds' collapsed limits as flat orbifolds.
problem Understanding collapsed limits of flat manifolds.
method Analyzing totally geodesic foliations and Gromov-Hausdorff limits.
result Identify collapsed limits as flat orbifolds and provide criteria for singularity.
Study provides limits on how surfaces can collapse.
problem Quantifying limits on surface collapse.
method Lower curvature bound and upper diameter bound.
result Quantitative obstruction to surface collapse.
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. Analyzes limits of flat metrics on complex curves.
problem Understanding limits of flat metrics on complex curves.
method Criterion based on piecewise affine weight function on intersection complex.
result Collapsed limits are metric graphs, non-collapsed are collections of curves.
Study Gromov-Hausdorff limits of K3 surface metrics via moduli compactification.
problem Understanding limits of K3 surface metrics.
method Moduli-theoretic framework for collapsing Ricci-flat Kahler metrics.
result Gromov-Hausdorff limits of hyperKahler metrics with fixed diameters.
Study limits of Kähler manifolds with bounded Ricci curvature.
problem Understanding the structure of limits of Kähler manifolds.
method Analysis of Gromov-Hausdorff limits with Ricci curvature bounds.
result Each tangent cone is homeomorphic to a normal affine variety.
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. Study shows how certain complex geometrical structures shrink to a simpler form.
problem Behavior of hyperkähler manifolds under specific conditions.
method Analyzes projective hyperkahler manifolds with holomorphic fibrations.
result Metrics of shrinking torus fibers collapse to a special Kahler manifold.
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.
Ancient solutions to Ricci flow on torus bundles have additional symmetries.
problem Understanding collapsed ancient solutions to the Ricci flow on compact manifolds.
method Algebraic and tameness assumptions on collapsing directions to prove additional torus symmetries.
result Ancient solutions to the Ricci flow on torus bundles converge to an Einstein metric on the base.
The study improves Perelman's theorems on Ricci flow.
problem No local collapse in Ricci flow on minimal projective manifolds.
method Localization of entropy functionals and further development of Li-Yau estimate.
result Generalization of no-local-collapsing theorem and pseudo-locality theorem.
Paper shows limits of Heisenberg manifolds are flat tori.
problem Understanding limits of sub-Riemannian Heisenberg manifolds.
method Analyzes collapsed Gromov--Hausdorff limits of compact Heisenberg manifolds.
result Collapsed limits are isometric to flat tori.
Study collapsing Calabi-Yau metrics and flows on fiber spaces.
problem Understanding the behavior of Calabi-Yau metrics and flows during collapsing.
method Analyzing the collapsing of Calabi-Yau metrics and Kähler-Ricci flows on fiber spaces.
result Identify the collapsed Gromov-Hausdorff limit and bounds for Hausdorff measure.
Researchers create metrics collapsing to a line, detailing near-fiber behavior.
problem Constructing metrics on elliptic K3 surfaces that collapse to a line.
method Family of collapsing Ricci-flat Kähler metrics with bounded curvatures.
result Precise description of metric degeneration near singular fibers.
This manuscript studies manifolds-with-boundary collapsing in the Gromov-Hausdorff topology. The main aim is an understanding of the relationship of the topology and geometry of a limiting sequence of manifolds-with-boundary to that of a limit space, which is presumed to be without geodesic terminals. The main result e…
Study on Kähler-Ricci flow on Fano bundles, showing fiber collapse and convergence.
problem Behavior of Kähler-Ricci flow on Fano bundles.
method Analysis of Kähler-Ricci flow on Fano bundles with specific fiber types and initial metrics.
result Fiber collapse and convergence to a metric on the base in Gromov-Hausdorff sense.
Study limits of curved spaces with boundaries.
problem Understanding geometric structures of curved spaces with boundary constraints.
method Gromov-Hausdorff convergence and collapsing analysis of compact Riemannian manifolds with boundary.
result Describe local geometric structure of limit spaces and establish stability results.
Continuity method on Fano fibrations converges to singular metrics.
problem Volume collapse of Kähler metrics on projective manifolds.
method Study of finite-time collapsing limits of the continuity method.
result Continuity method converges to singular Kähler metrics on the base in the weak sense.
Proves torus sequences can't collapse to intervals under curvature bounds.
problem Proving torus sequences can't collapse to intervals under curvature bounds.
method Contradiction proof using Yamaguchi fibration theorem and covers.
result Proves tori can't collapse to intervals under curvature constraints.
Study shows convergence of cscK surfaces in Hilbert scheme.
problem Understanding convergence of cscK surfaces.
method Gromov--Hausdorff convergence and Hilbert scheme approach.
result Established convergence of non-collapsed polarized cscK surfaces in a Hilbert scheme.
Existence and convergence of ancient Ricci flow solutions on compact homogeneous spaces.
problem Existence and characterization of ancient solutions to the Ricci flow on compact homogeneous spaces.
method General existence theorem and Gromov-Hausdorff convergence under rescaling.
result Convergence of collapsed ancient solutions to Einstein metrics on torus fibrations.
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. Paper compares Ricci flow volume changes and applies to Kähler-Ricci flow.
problem Volume comparison in Ricci flow and convergence of Kähler-Ricci flow.
method Derives a new relative volume comparison estimate and applies it to Kähler-Ricci flow.
result Generalizes Perelman's no local collapsing estimate and provides an analogue of Bishop-Gromov volume comparison for Ricci flow.
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.
The paper explores embedding Ricci flow solutions in flag manifolds.
problem Realizing Ricci flow solutions as embedded submanifolds.
method Investigation of invariant metrics in flag manifolds, proving global attractors and non-realizable collapses.
result Certain Ricci flow collapses cannot be embedded in Euclidean spaces.
Proves limits of Kahler-Einstein manifolds are smooth or orbifold.
problem Limits of Kahler-Einstein manifolds can be singular.
method Proves limits are either smooth or orbifold outside a subvariety.
result Non-collapsing and compact Gromov-Hausdorff limits are either smooth or orbifold.
Study of Calabi-Yau fibrations with restrictions on singularities.
problem Understanding the adiabatic behavior of Calabi-Yau metrics on fibrations.
method Develop techniques to study the limiting behavior of Calabi-Yau metrics, impose restrictions on singularity types, and prove uniform bounds.
result Uniform lower and upper bounds on the metric near singular fibres, and a uniform fibre diameter bound for a specific case.
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.
The flow contracts cone divisors on Kähler surfaces to points.
problem Analyzing the conical Kähler-Ricci flow on Hirzebruch surfaces.
method Using the momentum construction of Calabi, the conical Kähler-Ricci flow is studied on Hirzebruch surfaces.
result The flow either converges to a sphere or a single point, or contracts the cone divisor to a single point.
Study of Calabi-Yau metrics on converging manifolds, resolving conjectures.
problem Understanding Calabi-Yau metrics on converging manifolds.
method Analysis of Gromov-Hausdorff limits of metrics on Calabi-Yau fibrations.
result Gromov-Hausdorff limit is homeomorphic to the base of the fibration and discriminant locus has high Hausdorff codimension.
Extends arguments to limit structure in Calabi-Yau degenerations.
problem Understanding Gromov-Hausdorff limits in degenerating Calabi-Yau manifolds.
method Reduces conjecture to partial second-order estimate.
result Extends arguments to new settings.
Study of K3 surfaces' limits using moduli theory.
problem Understanding Gromov-Hausdorff limits of K3 surfaces.
method Explicit moduli-theoretic framework for Ricci-flat Kahler metrics.
result Proof of Kontsevich-Soibelman conjecture for K3 surfaces.
This paper is a sequel to arXiv:1108.0967. We further study Gromov-Hausdorff collapsing limits of Ricci-flat Kähler metrics on abelian fibered Calabi-Yau manifolds. Firstly, we show that in the same setup as arXiv:1108.0967, if the dimension of the base manifold is one, the limit metric space is homeomorphic to the bas…
Study of Chern-Ricci flow on Hopf surfaces, showing finite-time volume collapse and uniform bounds.
problem Understanding the Chern-Ricci flow on Hopf surfaces, especially minimal non-Kähler ones.
method Construction of locally conformally Kähler metrics and analysis of Chern-Ricci flow.
result Finite-time volume collapse and uniform upper bounds on the metric tensor.
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 estimates lead to Gromov-Hausdorff limits for Hermitian minimal models.
problem Uniform diameter and volume estimates for Chern-Ricci flow on Hermitian minimal models.
method Uniform diameter and volume estimates, local Kähler assumption, Perelman's reduced length, almost monotonicity formula for reduced volume.
result Gromov-Hausdorff convergence of the Chern-Ricci flow on Hermitian minimal models.
In the present paper, we determine the topologies of three-dimensional closed Alexandrov spaces which converge to lower dimensional spaces in the Gromov-Hausdorff topology.
Study on how Kähler-Einstein metrics behave during degenerations of manifolds.
problem Behavior of Kähler-Einstein metrics during degenerations of manifolds.
method Analysis of collapsing behavior of negative Kähler-Einstein metrics along degenerations of canonical polarized manifolds.
result Kähler-Einstein metrics on fibers collapse to a lower dimensional complete Riemannian manifold in the pointed Gromov-Hausdorff sense.
Eigenfunctions of collapsing Einstein manifolds are almost constant along fibers.
problem Eigenfunctions of collapsing Einstein manifolds.
method Cheeger-Colding's almost splitting theorem and harmonic almost splitting map.
result Eigenfunctions are almost constant along fibers in the L2-average sense. Study shows how 4D geometry becomes semiflat near curvature limits.
problem Understanding the geometry of almost Ricci-flat 4-manifolds.
method Analyzes Riemannian 4-manifolds converging to lower-dimensional limits with vanishing Ricci tensor.
result Semiflat Kaehler geometry emerges near curvature blowup regions in 4D space.
Study on topological properties and boundaries of RCD(K,N) spaces.
problem Understanding the topology and boundaries of non-collapsed RCD(K,N) spaces.
method Established topological regularity and stability, introduced boundary concept.
result Properties of boundaries and behavior under convergence studied.
Gravitational instantons collapse to a punctured plane with a special Kahler metric.
problem The collapse of gravitational instantons from a complex structure limit.
method Analysis of a sequence of ALH*-gravitational instantons and their collapse to a punctured plane.
result The moduli space of pointed ALH*-gravitational instantons collapses to a punctured plane with a special Kahler metric.
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.
We study unit horizontal bundles associated with Riemannian submersions. First we investigate metric properties of an arbitrary unit horizontal bundle equipped with a Riemannian metric of the Cheeger-Gromoll type. Next we examine it from the Gromov-Hausdorff convergence theory point of view, and we state a collapse the…
We study the limiting behavior of the Kahler-Ricci flow on P(OPn⊕OPn(−1)⊕(m+1)), assuming the initial metric satisfies the Calabi symmetry. We show that the flow either shrinks to a point, collapses to Pn or contracts a subvariety of c…
Uniform estimates prove convergence of Chern-Ricci flow on complex surfaces.
problem Proving convergence of Chern-Ricci flow on complex minimal surfaces.
method Uniform diameter estimates, volume non-collapsing estimates, Gromov-Hausdorff convergence; surface torsion estimate, uniform total variation bound, Green-weighted L^2 estimate, linear iteration of real Poisson equations.
result Uniform diameter estimates, volume non-collapsing estimates, Gromov-Hausdorff convergence for normalized Chern-Ricci flow on complex minimal surfaces.
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.