Study on collapsing Calabi-Yau manifolds and their metrics.
problem Understanding degenerations of Calabi-Yau manifolds with Ricci-flat Kahler metrics.
method Survey of recent developments, focusing on volume collapsing metrics.
result New insights into the behavior of Calabi-Yau manifolds under volume collapse.
Prove that collapsing CSC metrics can be perturbed to invariant collapsing CSC metrics.
problem Prove that collapsing constant scalar curvature metrics can be perturbed to invariant collapsing constant scalar curvature metrics.
method Prove that a sequence of constant scalar curvature metrics which is collapsing with bounded curvature to a manifold can be perturbed to a sequence of invariant collapsing constant scalar curvature metrics.
result Prove that a sequence of constant scalar curvature metrics which is collapsing with bounded curvature to a manifold can be perturbed to a sequence of invariant collapsing constant scalar curvature metrics.
New metric space concept and quasi-isometry properties explored.
problem Exploring new metric spaces and quasi-isometry properties.
method Introducing (b,c)-metric and defining collapsing maps.
result Collapsing maps preserve quasi-isometry properties.
Formula for scalar curvature under metric collapse.
problem Finding positive scalar curvature metrics.
method Formula involving scalar curvature and adapted orthonormal frame.
result Effect of metric collapse on scalar curvature.
Collapsibility is a combinatorial strengthening of contractibility. We relate this property to metric geometry by proving the collapsibility of any complex that is CAT(0) with a metric for which all vertex stars are convex. This strengthens and generalizes a result by Crowley. Further consequences of our work are: (1) …
Polyhedra collapse to subpolyhedra if they can be continuously shrunk onto them.
problem Characterizing when a polyhedron can be continuously shrunk onto a subpolyhedron.
method Piecewise-linear free deformation retraction and metric considerations.
result A polyhedron collapses to a subpolyhedron if and only if it admits a free deformation retraction onto that subpolyhedron.
In this paper, we study the collapsing behaviour of negative Kähler-Einstein metrics along degenerations of canonical polarized manifolds. We prove that for a toroidal degeneration of canonical polarized manifolds with the total space Q-factorial, the Kähler-Einstein metrics on fibers collapse to a lower dim…
New metrics prevent event collapse in contrast maximization frameworks.
problem Event collapse in contrast maximization frameworks.
method First principles of space-time deformation based on differential geometry and physics.
result Proposed metrics mitigate event collapse and do not harm well-posed warps.
The Kähler-Ricci flow on certain manifolds collapses to a canonical metric.
problem Understanding the behavior of Kähler-Ricci flow on compact manifolds.
method Asymptotic expansion of evolving metrics and analysis of the Iitaka fibration.
result The flow collapses to a canonical metric on the base of the Iitaka fibration.
Generalizes expansion and collapse theory to metric spaces.
problem Compactification of arbitrary metric spaces.
method Expands dual notions of expansion and collapse to arbitrary metric spaces and infinitely many moves.
result Proves compactification theorems, particularly for Z-set compactifications.
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.
I study Gromov-Hausdorff limits of complex curves endowed with singular flat metrics of constant diameter. I formulate a criterion that the limit is collapsed in terms of a certain piecewise affine weight function on the dual intersection complex of a semi-stable model of the degeneration introduced by Kontsevich and S…
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.
Study on K3 surfaces' collapsing and special Kähler structures.
problem Understanding the structure of K3 surfaces' collapsing metrics.
method Analyzing M2 and establishing connections to SKSs and Jacobian elliptic K3 surfaces. result Established a bijection between integral singular SKSs on P1 and Jacobian elliptic K3 surfaces. The goal of this paper is to study the stability of pure nilpotent structures on a manifold associated to different collapsed metrics. We prove that if two metrics on a n-manifold of bounded sectional curvature are L0-bi-Lipchitz equivalent and sufficient collapsed (depending on L0 and n), then up to a diffeo…
Gluing theorem for collapsing warped-QAC Calabi-Yau manifolds verified.
problem Behavior of warped-QAC Calabi-Yau metrics on affine quadrics.
method Gluing construction for collapsing warped-QAC Calabi-Yau manifolds.
result Verification of Yang Li's conjecture on warped QAC Calabi-Yau metrics.
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.
Two-dimensional collapsed spaces with lower Ricci bounds are topological surfaces.
problem Topology of collapsed spaces with lower Ricci bounds
method Prove that collapsed spaces are topological surfaces
result Collapsed spaces are topological surfaces
New metrics improve scRNA-seq perturbation modeling by reducing mode collapse.
problem Outperformed by simple mean prediction in scRNA-seq perturbation modeling.
method Introduce DEG-aware metrics (WMSE, Rw2(Δ)) and negative/positive baselines. result WMSE loss function reduces mode collapse and improves model performance.
Study collapsing geometry of hyperkähler 4-manifolds and prove conjectures.
problem Understanding collapsing geometry of hyperkähler 4-manifolds.
method Investigation of collapsing geometry and proving conjectures.
result Proved two conjectures about collapsed limits and asymptotic behavior of hyperkähler 4-manifolds.
We use the gluing method to give a refined description of the collapsing Calabi-Yau metrics on Calabi-Yau 3-folds admitting a Lefschetz K3 fibration.
Study collapsing geometry with Ricci curvature, proving Kähler metrics and Killing structures.
problem Collapsing geometry of Riemannian manifolds with Ricci curvature constraints.
method Locally bounded Ricci covering geometry and Ricci flow smoothing techniques.
result Volume collapsed Calabi-Yau manifolds admit Ricci-flat Kähler metrics and compatible Killing structures.
Einstein metrics are blocked by manifold features and group growth.
problem Existence of Einstein metrics on specific 4-manifolds.
method Analysis of collapsing and group growth effects.
result Several 4-manifolds cannot support Einstein metrics due to specific features.
For any elliptic K3 surface F:K→P1, we construct a family of collapsing Ricci-flat Kähler metrics such that curvatures are uniformly bounded away from singular fibers, and which Gromov-Hausdorff limit to P1 equipped with the McLean metric. There are well-known e…
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.
Study on energy of maps from K3 surface to flat orbifold.
problem Energy of maps from K3 surface to flat orbifold.
method Investigate Dirichlet energy of smooth maps and introduce an invariant.
result Ratio of energy to invariant converges to 1 for Foscolo's collapsing families.
Paper proves collapsing result for orbifolds without curvature bounds.
problem Proving collapsing result for orbifolds without curvature bounds.
method Introduces weak submersions and stratified Riemannian metrics.
result Allows Gromov-Hausdorff limits of orbifolds with strictly lower dimension.
The paper constructs gravitational instantons with unique collapse patterns.
problem Creating gravitational instantons with specific geometric properties.
method Using gluing construction, the metric collapses to quotients of R^3 with specific patterns.
result The constructed gravitational instantons have unique collapse patterns and exceptional points.
We construct large families of new collapsing hyperkähler metrics on the K3 surface. The limit space is the quotient of a flat 3-torus by an involution. Away from finitely many exceptional points the collapse occurs with bounded curvature. There are at most 24 exceptional points where the curvature concentrates, which …
We prove that the twisted Kahler-Einstein metrics that arise on the base of certain holomorphic fiber space with Calabi-Yau fibers have conical-type singularities along the discriminant locus. These fiber spaces arise naturally when studying the collapsing of Ricci-flat Kahler metrics on Calabi-Yau manifolds, and of th…
We exhibit families of Ricci-flat Kahler metrics on K3 surfaces which collapse to an interval, with Tian-Yau and Taub-NUT metrics occurring as bubbles. There is a corresponding continuous surjective map from the K3 surface to the interval, with regular fibers diffeomorphic to either 3-tori or Heisenberg nilmanifolds.
New metric properties show volume constraints in collapsing spaces.
problem Volume constraints in collapsing spaces.
method Generalization of recent progress in metric geometry involving the volume of balls of radius in a certain range with collapsing at different scales.
result For every Riemannian metric on a manifold of sufficiently small volume, there is a point with volume constraints in the universal cover.
The study defines a canonical nilpotent structure for certain collapsed manifolds.
problem Understanding the structure of collapsed Riemannian manifolds.
method Analyzes the nilpotent structure of manifolds with bounded Ricci curvature and Reifenberg local covering geometry.
result A canonical nilpotent structure can be defined and uniquely determined over regular limit spaces.
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.
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.
We establish a general "boundedness implies convergence" principle for a family of evolving Riemannian metrics. We then apply this principle to collapsing Calabi-Yau metrics and normalized Kähler-Ricci flows on torus fibered minimal models to obtain convergence results.
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.
We develop some techniques to study the adiabatic limiting behaviour of Calabi-Yau metrics on the total space of a fibration, and obtain strong control near the singular fibres by imposing restrictions on the singularity types. We prove a uniform lower bound on the metric up to the singular fibre, under fairly general …
We provide an algebraic description of the Teichmüller space and moduli space of flat metrics on a closed manifold or orbifold and study its boundary, which consists of (isometry classes of) flat orbifolds to which the original object may collapse. It is also shown that every closed flat orbifold can be obtained by col…
We study the collapsing behaviour of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold which admits an abelian fibration, when the volume of the fibers approaches zero. We show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature, and along the fibers the re…
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.
We provide a moduli-theoretic framework for the collapsing of Ricci-flat Kahler metrics via compactification of moduli varieties of Morgan-Shalen and Satake type. In patricular, we use it to study the Gromov-Hausdorff limits of hyperKahler metrics with fixed diameters, especially for K3 surfaces.
Study abelian varieties' Weil-Petersson metric asymptotics.
problem Asymptotic behavior of Weil-Petersson metric on abelian varieties.
method Linking asymptotic with multi-scale collapsing limits of parametrized flat tori.
result Refined description of Weil-Petersson metric on abelian varieties.
In this paper, we make progress on understanding the collapsing behavior of Calabi-Yau metrics on a degenerating family of polarized Calabi-Yau manifolds. In the case of a family of smooth Calabi-Yau hypersurfaces in projective space degenerating into the transversal union of two smooth Fano hypersurfaces in a generic …
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…
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.
Generalizes Rips' result on hyperbolic spaces to metric spaces, showing collapses for tree metrics.
problem Understanding the contractibility of Vietoris-Rips complexes in metric spaces.
method Extending Rips' result using geodesic defect and apparent pairs gradient.
result Vietoris-Rips complexes collapse to subforests for finite tree metrics.
New families of Ricci solitons found with collapsing volume.
problem Finding new Ricci solitons with specific volume behavior.
method Reduced soliton equation to Monge-Ampère equation coupled with ODEs.
result Explicit complete expanding solitons and existence results for other types.