Study Gromov-Hausdorff convergence of metric pairs and tuples.
problem Understanding convergence in metric spaces.
method Prove equivalence of definitions, embedding, completeness, and compactness theorems.
result Relative version of Fukaya's theorem and finiteness theorem for stratified spaces.
Study on convergence rate of Bergman metrics on Kähler manifolds.
problem Analyzing convergence rate of Bergman metrics on Kähler manifolds.
method Using Tian's peak section method to show uniform C1,α convergence. result Uniform C1,α convergence of Bergman metrics is demonstrated. Study approximates sub-Riemannian structures with Riemannian metrics and analyzes spectral convergence.
problem Approximating sub-Riemannian structures for analysis.
method Constructing Riemannian metrics tailored to sub-Riemannian structures and studying spectral convergence.
result Riemannian volumes converge to Popp's volume and spectral convergence of Laplace operators is studied.
New bounds on scalar curvature for metric sequences.
problem Bounding scalar curvature in metric sequences.
method Integral convergence of scalar curvature; point-wise scalar curvature lower bound.
result Limiting metric has scalar curvature lower bound.
Study on convergence of transformed metric spaces as dimensions grow.
problem Conditions for convergence of transformed metric spaces.
method Clarifying conditions for convergence of transformed spaces from original sequence and vice versa.
result Spheres and projective spaces converge to Gaussian space and its quotient as dimensions increase.
Study shows convergence of Lagrangian submanifolds under certain metrics.
problem Understanding convergence of Lagrangian submanifolds under specific metrics.
method Proves convergence to an embedded Lagrangian submanifold using a monotonicity lemma applied on a carefully-chosen metric ball.
result Convergence to an embedded Lagrangian submanifold implies convergence in the Hausdorff metric for a class of metrics.
The paper studies Kähler-Einstein metrics with singularities and their limits.
problem Analyzing Kähler-Einstein metrics with crossing edge singularities and their limits.
method Extending Guenancia's techniques, the paper shows convergence of metrics under specific angle conditions.
result Negatively curved Kähler-Einstein crossing edge metrics converge to mixed cusp and edge metrics smoothly away from the divisor.
We relate Lp convergence of metric tensors or volume convergence to a given smooth metric to Intrinsic Flat and Gromov-Hausdorff convergence for sequences of Riemannian manifolds. We present many examples of sequences of conformal metrics which demonstrate that these notions of convergence do not agree in general ev…
New metrics avoid high-dimensional analysis challenges, proving convergence without 'curse of dimensionality'.
problem High-dimensional analysis challenges in empirical measure convergence.
method Proposed a new class of probability metrics free of the curse of dimensionality.
result Convergence of empirical measures is free of the curse of dimensionality.
The study proves leafwise flat forms for Gauduchon metrics on Inoue-Bombieri surfaces.
problem Proving the existence of leafwise flat forms for Gauduchon metrics on Inoue-Bombieri surfaces.
method Using the ∂∂-class, the study proves the existence of leafwise flat forms for Gauduchon metrics on Inoue-Bombieri surfaces. result Uniform convergence of the normalized Chern-Ricci flow starting at any Gauduchon metric on all Inoue-Bombieri surfaces, with smooth convergence and bounded curvature for initial metrics in the ∂∂-class of the Tricerri/Vaisman metric. Ricci flow stability on manifolds with bounded geometry ensures convergence to hyperbolic metrics.
problem Stability and convergence of Ricci flow on manifolds with bounded geometry.
method Continuous dependence on initial conditions, sectoriality of Ricci-DeTurck flow generator, and Hölder norm analysis.
result Ricci flow converges to hyperbolic metrics under certain conditions.
Uniform convergence of metrics on surfaces with bounded curvature measures proved.
problem Proving uniform convergence of metrics on Alexandrov surfaces with bounded integral curvature.
method Weak convergence of measures and analytic approximation of metrics.
result Uniform convergence of metrics on Alexandrov surfaces proved.
New Calabi-Yau metrics converge polynomially to Calabi model space.
problem Finding complete Calabi-Yau metrics with polynomial convergence rate.
method Defined new metrics on Calabi-Yau complements with ample normal bundles.
result Uniqueness of these metrics within a cohomology class.
This is an intuitive survey of extrinsic and intrinsic notions of convergence of manifolds complete with pictures of key examples and a discussion of the properties associated with each notion. We begin with a description of three extrinsic notions which have been applied to study sequences of submanifolds in Euclidean…
Study group actions in metric spaces, proving convergence of lens spaces.
problem Understanding convergence in metric measure spaces with group actions.
method Generalized box and observable distances, applied mass-transport theory.
result Sequence of lens spaces converging to infinite-dimensional complex projective space.
Study shows convergence of Fubini-Study currents to equilibrium metrics on Kähler manifolds.
problem Convergence of Fubini-Study currents to equilibrium metrics in Kähler geometry.
method Analysis of continuous Hermitian metrics and their Fubini-Study currents on line bundles.
result The scaled difference between Fubini-Study currents and equilibrium metrics converges to zero in the sense of currents.
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.
Study on Kähler-Einstein metrics with polynomial convergence rates.
problem Understanding convergence rates of singular Kähler-Einstein metrics.
method Analyzing non-collapsed limits and tangent cones of polarized Kähler-Einstein manifolds.
result Polynomial convergence of Kähler potentials on tangent cones.
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.
Holonomy groups of metric connections converge in a monotonic way.
problem Monotonicity of holonomy groups under convergence of metric connections.
method Proving the monotonicity of holonomy groups for sequences of metric connections converging in C0. result The holonomy group of the limit connection is contained in the holonomy group of the initial connections.
Study confirms conjecture about Kähler metrics on smooth minimal models.
problem Behavior of constant scalar curvature Kähler metrics on smooth minimal models.
method Analysis of metrics in a neighborhood of the canonical class.
result Convergence to singular Kähler Einstein metric in the canonical class.
Study shows tori metrics converging to flat under specific conditions.
problem Understanding convergence of metrics on tori with non-negative scalar curvature.
method Uniformly conformal metrics and controlled geometry sequences.
result Sequence of metrics converges to flat metric in multiple senses.
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. 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.
Study sequences of static spacetimes using null distance convergence.
problem How to define convergence for sequences of spacetimes.
method Define null distance metric space structure compatible with Lorentzian structure.
result Prove VADB theorem for sequences of static spacetimes with null distance.
We introduce a natural definition of Lp-convergence of maps, p≥1, in the case where the domain is a convergent sequence of measured metric space with respect to the measured Gromov-Hausdorff topology and the target is a Gromov-Hausdorff convergent sequence. With the Lp-convergence, we establish a theory of …
The Hesse-Koszul flow converges to the Hesse-Einstein metric on compact Hessian manifolds.
problem Existence and convergence of Hesse-Einstein metrics on compact Hessian manifolds.
method Study of the Hesse-Koszul flow and its convergence properties.
result The flow converges to the unique Hesse-Einstein metric under certain conditions.
The paper examines conditions for Gromov-Hausdorff convergence of metric quotients and provides examples of conic-flat surfaces.
problem Conditions for Gromov-Hausdorff convergence of metric quotients.
method Analyzes sufficient conditions for Gromov-Hausdorff convergence of metric quotients of a metric space.
result Concrete examples of sequences of two-dimensional conic-flat spheres converging to spheres with singularities.
Study shows convergence of anticanonically balanced metrics to Kähler-Einstein metrics on Fano manifolds.
problem Finding anticanonically balanced metrics on Fano manifolds.
method Simplification of Donaldson's proof using Berezin-Toeplitz quantization.
result Sequence of anticanonically balanced metrics converges to Kähler-Einstein metric.
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.
Proves compactness for timed-metric spaces using new distance and maps.
problem Weak convergence of space-times using timed-Hausdorff distance.
method Uses Gromov's original compactness theorem and introduces addresses.
result Establishes compactness theorem for intrinsic timed-Hausdorff convergence.
We study the quantization of coupled Kähler-Einstein (CKE) metrics, namely we approximate CKE metrics by means of the canonical Bergman metrics, so called the ``balanced metrics''. We prove the existence and weak convergence of balanced metrics for the negative first Chern class, while for the positive first Chern clas…
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.
Ricci flow converges to Taub-NUT metric under specific conditions.
problem Analyzing convergence of Ricci flow solutions to Taub-NUT metric.
method Study of Ricci flow starting from a specific metric on R4. result Ricci flow converges to Taub-NUT metric in infinite time under certain conditions.
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 on convergence rate of weighted Yamabe flow.
problem Weighted Yamabe problem on smooth metric measure spaces.
method Weighted Yamabe flow and its convergence rate analysis.
result Study and analysis of convergence rate of the weighted Yamabe flow.
We characterize the rate of convergence of a converging volume-normalized Yamabe flow in terms of Morse theoretic properties of the limiting metric. If the limiting metric is an integrable critical point for the Yamabe functional (for example, this holds when the critical point is non-degenerate), then we show that the…
A comparison theorem for the isoperimetric profile on the universal cover of surfaces evolving by normalised Ricci flow is proven. For any initial metric, a model comparison is constructed that initially lies below the profile of the initial metric and which converges to the profile of the constant curvature metric. Th…
In this work, convergence of evolving Finslerian metrics first in a general flow next under Finslerian Ricci flow is studied. More intuitively it is proved that a family of Finslerian metrics g(t) which are solutions to the Finslerian Ricci flow converge in C∞ to a smooth limit Finslerian metric as t ap…
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.
In this paper we will discuss local coordinates canonically corresponding to a Kahler metric. We will also discuss and prove the C∞ convergence of Bergman metrics following Tian's result on C2 convergence of Bergman metrics. At the end, we present an interesting characterization of ample line bundle that cou…
We investigate the Kahler-Ricci flow on holomorphic fiber spaces whose generic fiber is a Calabi-Yau manifold. We establish uniform metric convergence to a metric on the base, away from the singular fibers, and show that the rescaled metrics on the fibers converge to Ricci-flat Kahler metrics. This strengthens previous…
In a recent paper Donaldson defines three operators on a space of Hermitian metrics on a complex projective manifold: T,Tν,TK. Iterations of these operators converge to balanced metrics, and these themselves approximate constant scalar curvature metrics. In this paper we investigate the convergence properties of …
The paper studies convergence of cosmological spacetimes using null distance.
problem Convergence of cosmological spacetimes with compact slices.
method Using null distance and Gromov-Hausdorff convergence, the paper establishes convergence results for spacetimes with mild extension properties.
result Uniform convergence of null distances and Gromov-Hausdorff convergence for monotone sequences of spacetimes.
Study Kähler metrics with constant scalar curvature using coupled equations.
problem Finding Kähler metrics with constant scalar curvature.
method Solving a system of elliptic equations for a Kähler metric and a closed (1,1)-form, proving higher order estimates and smooth convergence.
result Smooth convergence to a cscK metric coupled to a harmonic (1,1)-form under uniform estimates.
New approach to geometric quantization for symplectic manifolds.
problem Quantization of symplectic manifolds with non-singular Lagrangian fibrations.
method Using spectral convergence of metric measure spaces, the authors develop a new geometric quantization approach.
result Spectral and quantum Hilbert space convergence results for Kähler and almost Kähler quantizations.
The Yamabe flow on flat manifolds converges to a scalar flat metric.
problem Analyzing the convergence of Yamabe flow on asymptotically flat manifolds.
method Yamabe flow starting from an asymptotically flat manifold, convergence analysis.
result The flow converges to an asymptotically flat, scalar flat metric under certain conditions.
The paper studies the convergence of harmonic metrics on Higgs bundles.
problem Analyzing the asymptotic behavior of harmonic metrics on Higgs bundles.
method Investigates the convergence of harmonic metrics on stable Higgs bundles of degree 0.
result The sequence of harmonic metrics converges to a decoupled harmonic metric at an exponential rate.