Study shows limits of metrics with positive scalar curvature on spheres.
problem Non-negativity of scalar curvature is not preserved under certain limits.
method Examined metrics conformal to the round metric on Sn for n≥4. result Any conformal metric to the round metric on Sn for n≥4 can be a limit of metrics with positive scalar curvature. Study of limits of Einstein-Bogomol'nyi metrics on P^1 in two regimes.
problem Understanding limits of Einstein-Bogomol'nyi metrics on P^1.
method Analysis of two regimes: dissolving limit and large volume limit.
result Recovery of Einstein-Bogomol'nyi metrics on C with total string number N' for each N'.
Central limit theorem for Green metrics on hyperbolic groups.
problem Proving a central limit theorem for Green metrics on hyperbolic groups.
method Proving a central limit theorem for Green metrics on hyperbolic groups using probability measures and ordering elements.
result Proved a central limit theorem for Green metrics on hyperbolic groups.
Study of Calabi-Yau manifold degenerations near complex structure limits.
problem Understanding polarized degenerations of Calabi-Yau manifolds.
method Improvement of metric convergence results on generic regions.
result Metric convergence for collapsing Ricci-flat Kähler metrics on generic regions.
Uniform convergence of metrics on vortex moduli space in Bradlow limit.
problem Understanding the geometry of vortex moduli spaces.
method Proof of uniform convergence of metrics using normalized L2 metric and Fubini-Study metric. result Establishes the Fubini-Study metric as the limit of the normalized L2 metric in the Bradlow limit. We study the behaviour of families of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold when the Kahler classes degenerate to the boundary of the ample cone. We prove that if the limit class is big and nef the Ricci-flat metrics converge smoothly on compact sets outside a subvariety to a limit incomplete Ri…
The paper examines sequences of metric spaces converging to compact limits with specific properties.
problem Understanding convergence of metric spaces with compact limits.
method Analyzes sequences of metric spaces with increasing distance functions and uniform bounds, proving convergence under certain conditions.
result Uniform and Gromov-Hausdorff convergence and volume preserving intrinsic flat convergence to compact limits.
This is a continuation of paper \cite{Li}. On any toric Fano manifold, we discuss the behavior of limit metric of a sequence of metrics, which are solutions to a continuity family of complex Monge-Ampere equations in Kahler-Einstein problem. We show that the limit metric satisfies a singular complex Monge-Ampere equati…
Newly discovered Eguchi-Hanson metric arises from edge metrics.
problem Understanding limits of compact singular Einstein spaces.
method Constructing Kahler-Einstein edge metrics on Calabi-Hirzebruch manifolds.
result Eguchi-Hanson metric emerges as a Gromov-Hausdorff limit.
The paper studies scaling limits of Wasserstein metrics on Gaussian mixture models.
problem Understanding the scaling limits of Wasserstein metrics on Gaussian mixture models.
method Scaling limit approach on Gaussian mixture models, including inhomogeneous and extended models.
result Existence of the limit of the Wasserstein metric after renormalization for GMMs with zero variance.
Proves limit curve theorem for incomplete metric spaces, applies to null distance in Lorentzian manifolds.
problem Control of Lorentzian lengths of limit curves in incomplete metric spaces.
method Proves limit curve theorem for incomplete metric spaces and applies to null distance.
result Strong control on Lorentzian lengths of limit curves in Sormani and Vegas' null distance.
Study noncollapsed F-limit metric solitons, proving properties similar to smooth Ricci shrinkers.
problem Understanding noncollapsed F-limit metric solitons in Ricci flow.
method Systematic study and proving properties similar to smooth Ricci shrinkers.
result Proves quadratic lower bound for scalar curvature, local gap theorem, global Sobolev inequality, and optimal volume growth lower bound.
This is the second of a series of three papers which provide proofs of results announced in arXiv:1210.7494. In this paper we consider the Gromov-Hausdorff limits of metrics with cone singularities in the case when the limiting cone angle is less than 2π. We show that these are in a natrual way projective algebraic var…
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.
The study examines conical Kähler-Einstein metrics on specific Fano manifolds and their behavior at the limit angle.
problem Analyzing conical Kähler-Einstein metrics on rank one horosymmetric Fano manifolds.
method Investigates the convergence of these metrics as the cone angle approaches zero, focusing on the behavior on the complement of a codimension one orbit.
result The metrics converge to the Kähler-Einstein metric on the basis and Stenzel's Ricci flat Kähler metrics on the fibers.
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…
Study smooth convergence of metric flows from F-limits.
problem Smooth convergence of F-limit flows. method Extensively studied metric flows and F-limits, showing smooth convergence at regular points. result Each regular point on the limit is a point of smooth convergence.
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.
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…
Finite approximations help reconstruct countable metric and ultrametric spaces.
problem Reconstructing countable metric and ultrametric spaces.
method Topological reconstruction using inverse limits of finite T0 spaces. result Countable metric and ultrametric spaces can be reconstructed as finite approximations.
The abstract discusses how Kähler-Einstein metrics relate to algebraic geometry.
problem Understanding the connection between Kähler-Einstein metrics and algebraic geometry.
method Exploring metric limits and rescalings of Kähler-Einstein metrics in relation to moduli spaces and singularities.
result Proposes tentative conjectural pictures connecting Kähler-Einstein metrics and algebraic geometry.
In the paper arXiv:1411.4887 [math.AP] it is shown that the set of Riemannian metrics which do not admit global limiting Carleman weights is open and dense, by studying the conformally invariant Weyl and Cotton tensors. In the paper arXiv:1011.2507 [math.DG] it is shown that the set of Riemannian metrics which do not a…
A W1,p-metric on an n-dimensional closed Riemannian manifold naturally induces a distance function, provided p is sufficiently close to n. If a sequence of metrics gk converges in W1,p to a limit metric g, then the corresponding distance functions dgk subconverge to a limit distance function …
Paper studies convergence of nonnegative scalar curvature metrics to a specific limit space.
problem Understanding convergence of nonnegative scalar curvature metrics.
method Analyzes a sequence of warped product metrics on $\Sph^2 imes \Sph^1$.
result Sequence converges to an extreme limit space in specific senses.
We consider a notion of balanced metrics for triples (X,L,E) which depend on a parameter α, where X is smooth complex manifold with an ample line bundle L and E is a holomorphic vector bundle over X. For generic choice of α, we prove that the limit of a convergent sequence of balanced metrics leads to a Hermitian-Einst…
Study geodesics on K3 surfaces near orbifold limit.
problem Understanding geodesics on K3 surfaces near the orbifold limit.
method Improves metric estimates for K3 surfaces, uses hyperkähler identities.
result Restrictions and existence conditions for stable geodesics.
Deviation inequalities and limit laws for random walks on metric spaces.
problem Understanding random walks on metric spaces with contracting isometries.
method Adapting Gouëzel's pivotal time construction to establish deviation inequalities.
result Exponential bounds and limit laws for random walks on mapping class groups and CAT(0) spaces.
Study geodesic Lie groups' convergence to limits with quantitative estimates.
problem Quantifying convergence rates of geodesic Lie groups to their limits.
method Estimates on the difference between original metrics and asymptotic/tangent metrics.
result Sharpens existing bounds on convergence rates.
Motivated by the classical statements of Mirror Symmetry, we study certain Kahler metrics on the complexified Kahler cone of a Calabi-Yau threefold, conjecturally corresponding to approximations to the Weil-Petersson metric near large complex structure limit for the mirror. In particular, the naturally defined Riemanni…
We generalize the observable diameter and the separation distance for metric measure spaces to those for pyramids, and prove some limit formulas for these invariants for a convergent sequence of pyramids. We obtain various applications of our limit formulas as follows. We have a criterion of the phase transition proper…
Paper generalizes balanced metrics existence to singular cases using Quot-scheme limit.
problem Existence of balanced metrics and Gieseker stability of vector bundles.
method Quot-scheme limit of Fubini-Study metrics and Bergman 1-parameter subgroups.
result Existence of balanced metrics equivalent to Gieseker stability for singular cases.
New metrics link Kähler and pp-wave spacetimes.
problem Connecting Kähler and pp-wave spacetimes.
method Constructing families of complete almost Kähler metrics by deforming pp-waves.
result Established a one-to-one correspondence between almost Kähler metrics and pp-wave spacetimes.
This is the third and final paper in a series which establish results announced in arXiv:1210.7494. In this paper we consider the Gromov-Hausdorff limits of metrics with cone singularities in the case when the limiting cone angle approaches 2π. We also put all our technical results together to complete the proof of the…
Study the limit of Calabi-Yau metrics with degenerate skeletons.
problem Understanding the behavior of Calabi-Yau metrics with degenerate skeletons.
method Using polarised degenerations and optimal transport problems.
result Describe the limiting behaviour of the Calabi-Yau potential.
Researchers prove existence of Kähler-Einstein metrics with conic singularities on Fano manifolds.
problem Existence of Kähler-Einstein metrics with conic singularities on Fano manifolds.
method Analyzing automorphisms and limits at various scales to prove the existence of metrics.
result Existence of Kähler-Einstein metrics with conic singularities for β>β∗ close to β∗. 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.
Study shows Kähler-Einstein metric singularities linked to curvature.
problem Understanding singularities of Kähler-Einstein metrics.
method Relates singularities to holomorphic sectional curvature of conical geometry.
result Provides second-order estimates with explicit constants.
The paper characterizes limits of manifolds using Gromov-Hausdorff metric.
problem Characterizing limits of generalized manifolds using Gromov-Hausdorff metric.
method Using Gromov-Hausdorff metric dG, the paper proves that manifold-like generalized n-manifolds are limits of topological n-manifolds under certain conditions. result Manifold-like generalized n-manifolds are limits of topological n-manifolds under specific conditions. 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…
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.
New metrics derived from geodesics simplify semi-Riemannian geometry.
problem Simplifying complex semi-Riemannian metrics.
method Constructing plane wave limits along geodesics.
result Generalizes Penrose's limit and encodes tensorial geometry.
We study the limits of sequences of spheres and complex projective spaces with unbounded dimensions. A sequence of spheres (resp. complex projective spaces) either is a Levy family, infinitely dissipates, or converges to (resp. the Hopf quotient of) a virtual infinite-dimensional Gaussian space, depending on the size o…
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.
Study higher rank inner products and their tilings to describe tori degenerations.
problem Understanding metric degenerations of tori.
method Introduce higher rank inner products and their tilings, use to describe degenerations.
result Describe metric degenerations of polarized tori and Hausdorff limits of tilings.
In this paper, we study the weak compactness of the set of conformal metrics in any Riemann surface without boundary whose Calabi energy and area are uniformly bounded. We prove that for any sequence of such metrics, there alwasy exists a subsequence which converges in H\sp{2,2}_\sb{loc} everywhere except a finite numb…
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.
We investigate various limits of the twistor spaces associated to the self-dual metrics on n CP ^2, the connected sum of the complex projective planes, constructed by C. LeBrun. In particular, we explicitly present the following 3 kinds of degenerations whose limits of the metrics are: (a) LeBrun metrics on (n-1) CP ^2…
We study adiabatic limits of Ricci-flat Kahler metrics on a Calabi-Yau manifold which is the total space of a holomorphic fibration when the volume of the fibers goes to zero. By establishing some new a priori estimates for the relevant complex Monge-Ampere equation, we show that the Ricci-flat metrics collapse (away f…