Study on the limits of projective special real manifolds and their symmetries.
problem Understanding the limits of projective special real manifolds.
method Evolution of defining polynomial and centro-affine fundamental form along curves.
result Found a list of possible limit geometries and a lower bound for symmetry groups.
This paper is devoted to the framework of direct limit of anchored Banach bundles over a convenient manifold which is a direct limit of Banach manifold. In particular we give a criterion of integrability for distributions on such convenient manifolds which are locally direct limits of particular sequences of Banach anc…
Study proves uniqueness of asymptotic limits for specific manifolds.
problem Proving uniqueness of asymptotic limits for Ricci-flat manifolds with linear volume growth.
method Established using natural curvature and cross section assumptions.
result Uniqueness and exponential convergence rate for complete noncollapsed Ricci-flat manifolds with linear volume growth.
Study on the geometry of limit spaces of manifolds with boundary.
problem Understanding the geometry of limit spaces of manifolds with boundary.
method Developed infinitesimal geometry for limit spaces under curvature and diameter bounds.
result Determined the infinitesimal structure and Hausdorff dimensions of boundary singular sets.
Integrability criterion for projective limits of Banach distributions on Fréchet manifolds.
problem Integrability of projective limits of involutive bundles on Banach manifolds.
method An integrability criterion for a projective limit of Banach distributions.
result Result of integrability of projective limit of involutive bundles on a projective sequence of Banach manifolds.
Smooth deformation of Moishezon manifolds preserves their Moishezon property.
problem Preserving Moishezon property under smooth deformation.
method Smooth deformation over a unit disk in C.
result Deformation limit of Moishezon manifolds is Moishezon.
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.
The paper connects geodesic flows and limit sets on visibility manifolds.
problem Understanding dynamics and ergodic properties on non-compact visibility manifolds.
method Analyzing geodesic flows and Patterson-Sullivan measures on visibility manifolds without conjugate points.
result The positivity of the Patterson-Sullivan measure of the Myrberg limit set is equivalent to the conservativity of the geodesic flow.
Uniqueness of asymptotic limits for Ricci-flat manifolds with linear volume growth is proven.
problem Proving uniqueness of asymptotic limits for noncollapsed Ricci flat manifolds with linear volume growth.
method Relating uniqueness to the existence of a harmonic function asymptotic to a Busemann function, proving uniqueness via a monotone quantity.
result Proves uniqueness of the asymptotic limit and establishes a polynomial convergence rate.
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.
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. 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.
The paper studies symplectic forms on projective limits of Banach bundles and their Darboux Theorem.
problem Conditions for weak symplectic forms on projective limits of Banach bundles.
method Analyzing projective sequences of Banach bundles and applying Darboux Theorem.
result Necessary and sufficient conditions for the Darboux Theorem on projective limits of Banach manifolds.
We discuss the Euclidean limit of hyperbolic SU(2)-monopoles, framed at infinity, from the point of view of pluricomplex geometry. More generally, we discuss the geometry of hypercomplex manifolds arising as limits of pluricomplex manifolds.
Paper studies limits of Kähler-Ricci flow on Fano G-manifolds.
problem Analyzing the limits of Kähler-Ricci flow on Fano G-manifolds.
method Proves the Gromov-Hausdorff limit of Kähler-Ricci flow on Fano G-manifolds is a horosymmetric variety.
result The limit of Kähler-Ricci flow on Fano G-manifolds is a horosymmetric variety.
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.
Triangle comparison for Kaehler manifolds with curvature bounds.
problem Understanding curvature bounds in Kaehler manifolds and their limits.
method Analog of triangle comparison for Kaehler manifolds with holomorphic bisectional curvature.
result Curvature bounds pass to noncollapsed Gromov-Hausdorff limits.
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.
In this paper, we will show that the limit of some quasilocal mass integrals of the coordinate spheres in an asymptotically hyperbolic (AH) manifold is the mass integral of the AH manifold. This is the analogue of the well known result that the limit of the Brown-York mass of coordinate spheres is the ADM mass in an as…
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.
In this paper we produce a sequence of Riemannian manifolds Mjm, m≥2, which converge in the intrinsic flat sense to the unit m-sphere with the restricted Euclidean distance. This limit space has no geodesics achieving the distances between points, exhibiting previously unknown behavior of intrinsic flat lim…
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. In this paper we continue to study Gromov-Hausdorff limits of Kahler manifolds and algebraic geometry. Our main focus is on the algebro-geometric meaning of Riemannian tangent cones and rescaled limits.
The aim of the paper is to investigate the relation between inverse limit of branched manifolds and codimension zero laminations. We give necessary and sufficient conditions for such an inverse limit to be a lamination. We also show that codimension zero laminations are inverse limits of branched manifolds. The inverse…
We study non-collapsed Gromov-Hausdorff limits of Kähler manifolds with Ricci curvature bounded below. Our main result is that each tangent cone is homeomorphic to a normal affine variety. This extends a result of Donaldson-Sun, who considered non-collapsed limits of polarized Kähler manifolds with two-sided Ricci curv…
Limits of manifolds with Kato bound on Ricci curvature are rectifiable.
problem Understanding limits of manifolds with specific curvature bounds.
method Proving rectifiability of limits of manifolds with Kato bound on Ricci curvature.
result Rectifiability of limits of manifolds with Kato bound on Ricci curvature.
Study of large mass limits of G2 and Calabi-Yau monopoles on specific manifolds.
problem Understanding the behavior of monopoles in the large mass limit on G2 and Calabi-Yau manifolds.
method Developed a structure theory for the limit of SU(2) G2-monopoles and Calabi-Yau monopoles, extracting singular abelian G2-monopoles with Dirac singularities. result Proved an energy identity for monopole bubbles in the large mass limit.
The paper proves a conjecture about manifold limits and characterizes their structure.
problem Characterizing limits of manifolds with a uniform contractibility function.
method Short proof using Gromov-Hausdorff distance and ANR properties.
result Obstruction vanishes if and only if the manifold can be approximated by PL-manifolds.
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. In this paper, we study the structure of the limit space of a sequence of almost Einstein manifolds, which are generalizations of Einstein manifolds. Roughly speaking, such manifolds are the initial manifolds of some normalized Ricci flows whose scalar curvatures are almost constants over space-time in the L1-sense,…
This paper examines limits of Riemannian 2-manifolds with bounded curvature.
problem Understanding the limits of Riemannian 2-manifolds with bounded curvature.
method Uniform semi-locally 1-connected sequences of closed connected Riemannian 2-manifolds with bounded total absolute curvature.
result Description of Gromov-Hausdorff limits of the sequences.
Proves CLT for Brownian paths on pinched negative curvature manifolds.
problem Distribution of Brownian paths on pinched negative curvature manifolds.
method Proof of central limit theorem for distances and Green functions.
result Central limit theorem holds for Brownian paths in pinched negative curvature.
This paper studies limits of aspherical manifolds with specific curvature conditions.
problem Understanding the Gromov-Hausdorff limits of aspherical manifolds with given curvature constraints.
method Analyzing sequences of compact manifolds with Ricci curvature or sectional curvature conditions, and using diffeomorphism or homeomorphism properties.
result If the manifolds are diffeomorphic or homeomorphic to nilmanifolds, their limits are also diffeomorphic or homeomorphic to nilmanifolds.
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.
We investigate the differential calculus defined by Ashtekar and Lewandowski on projective limits of manifolds by means of cylindrical smooth functions and compare it with the C^infty calculus proposed by Froehlicher and Kriegl in more general context. For products of connected manifolds, a Boman theorem is proved, sho…
Study adiabatic limits of Calderon projector on manifolds with cylindrical ends.
problem Constructing Calderon projector in a specific geometric setting.
method Using resolvent estimates and b-calculus framework.
result Extend a result on adiabatic limits of Cauchy data spaces.
New manifolds with negative curvature limit to one with negative curvature.
problem Preserving nonnegative scalar curvature under intrinsic flat convergence.
method Constructing sequences of manifolds with positive scalar curvature.
result Intrinsic flat limit of manifolds with negative scalar curvature.
Study shows different fundamental groups for manifolds with same limit.
problem Understanding fundamental groups of manifolds with non-negative Ricci curvature.
method Constructed sequences of manifolds with specific properties.
result Found manifolds with same limit but different fundamental groups.
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. Constructs knots from 3-manifolds with specified geometric limits.
problem Build knots from 3-manifolds with specific geometric properties.
method Constructs explicit families of knots converging to specified 3-manifolds.
result Constructs knots from geometrically finite 3-manifolds with one end.
Study on systole of random hyperbolic 3-manifolds, proving limit exists and calculating it.
problem Understanding the systole of random hyperbolic 3-manifolds.
method Modeling random hyperbolic 3-manifolds using truncated tetrahedra, calculating expected systole limit as volume increases.
result Closed formula and numerical approximation for the limit of the expected systole as volume tends to infinity.
We obtain a Central Limit Theorem for closed Riemannian manifolds, clarifying along the way the geometric meaning of some of the hypotheses in Bhattacharya and Lin's Omnibus Central Limit Theorem for Fréchet means. We obtain our CLT assuming certain stability hypothesis for the cut locus, which always holds when the ma…
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…
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…
The purpose of this paper is to study the property of the resolvent of the Laplace-Beltrami operator on a noncompact complete Riemannian manifold with various ends each of which has a different limit of the growth rate of the Riemannian measure at infinity, in particular, focusing on the limiting absorption principle. …
We present some recent results on the behavior of the spectrum of the differential form Laplacian on a Riemannian foliated manifold when the metric on the ambient manifold is blown up in directions normal to the leaves (in the adiabatic limit).
Study inradius collapsed manifolds with lower Ricci curvature bounds, proving properties of their limits.
problem Characterizing limits of inradius collapsed manifolds with lower Ricci curvature bounds.
method Analyzing families of manifolds with specific curvature and boundary conditions, proving properties of the limits.
result Limits of inradius collapsed manifolds have at most two boundary components and a lower Ricci curvature bound.
Defines and extends Lie algebroid prolongations in convenient settings.
problem Adapting Lie algebroid prolongations to convenient settings.
method Defined and adapted Lie algebroid prolongations over fibred manifolds.
result Stability of prolongations under projective and direct limits.