A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Lower Ricci curvature bound prevents first Betti number from dropping more than dimension in collapsing manifolds.
In this paper, we study collapsed manifolds with boundary, where we assume a lower sectional curvature bound, two sides bounds on the second fundamental forms of boundaries and upper diameter bound. Our main concern is the case when inradii of manifolds converge to zero. This is a typical case of collapsing manifolds w…
We will simplify the earlier proofs of Perelman's collapsing theorem of 3-manifolds given by Shioya-Yamaguchi and Morgan-Tian. Among other things, we use Perelman's semi-convex analysis of distance functions to construct the desired local Seifert fibration structure on collapsed 3-manifolds. The verification of Perelma…
Study collapsing geometry of hyperkähler 4-manifolds and prove conjectures.
We study collapsed manifolds with Ricci bounded covering geometry i.e., Ricci curvature is bounded below and the Riemannian universal cover is non-collapsed or consists of uniform Reifenberg points. Via Ricci flows' techniques, we partially extend the nilpotent structural results of Cheeger-Fukaya-Gromov, on collapsed …
Survey on collapsing manifolds using group actions and foliations.
Gluing theorem for collapsing warped-QAC Calabi-Yau manifolds verified.
We will simplify earlier proofs of Perelman's collapsing theorem for 3-manifolds given by Shioya-Yamaguchi and Morgan-Tian. Among other things, we use Perelman's critical point theory (e.g., multiple conic singularity theory and his fibration theory) for Alexandrov spaces to construct the desired local Seifert fibratio…
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…
Study inradius collapsed manifolds with lower Ricci curvature bounds, proving properties of their limits.
In the last two decades, one of the most important developments in Riemannian geometry is the collapsing theory of Cheeger-Fukaya-Gromov. A Riemannian manifold is called (sufficiently) collapsed if its dimension looks smaller than its actual dimension while its sectional curvature remains bounded (say a very thin flat …
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 -factorial, the Kähler-Einstein metrics on fibers collapse to a lower dim…
Ricci flow smooths locally collapsing manifolds with controlled curvature.
In this paper, the relationship between the existence of special lagrangian submanifolds and the collapsing of Calabi-Yau manifolds is studied. First, special lagrangian fibrations are constructed on some regions of bounded curvature and sufficiently collapsed in Ricci-flat Calabi-Yau manifolds. Then, in the opposite d…
This is an expositiry article on collapsing theory in Riemannian geometry written for the Modern Encyclopedia of Mathematical Physics (MEMPhys). We focus on describing the geometric and topological structure of collapsed/non-collapsed regions in Riemannian manifold under various curvature assumptions. Numerous applicat…
Study shows properties of Gromov-Hausdorff limit of frame bundles for non-collapsed manifolds.
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) …
Formula for scalar curvature under metric collapse.
Study collapsing geometry with Ricci curvature, proving Kähler metrics and Killing structures.
We study relations between certain totally geodesic foliations of a closed flat manifold and its collapsed Gromov-Hausdorff limits. Our main results explicitly identify such collapsed limits as flat orbifolds, and provide algebraic and geometric criteria to determine whether they are singular.
Paper shows limits of Heisenberg manifolds are flat tori.
New collapsing mechanism for G2-manifolds discovered.
Ancient Ricci flows on non-collapsed manifolds have finite fundamental groups.
New examples of manifolds with lower scalar curvature bounds and submanifold collapse.
Einstein metrics are blocked by manifold features and group growth.
We prove that a 3-dimensional compact Riemannian manifold which is locally collapsed, with respect to a lower curvature bound, is a graph manifold. This theorem was stated by Perelman and was used in his proof of the geometrization conjecture.
This short note studies the collapsing behavior of the Kähler-Ricci flow on a compact Kähler manifold X admitting a holomorphic submersion X -> B where B is a Kähler manifold of lower dimension than X. We give cohomological and curvature conditions under which the fibers collapse at the optimal rate ~(T-t)^{1/2}
The paper analyzes graph Laplacians on manifolds with curvature bounds and applies to non-collapsed spaces.
Aspherical manifolds with bounded curvature have non-trivial abelian subgroups in their fundamental groups.
We survey some recent developments on the problem of understanding degenerations of Calabi-Yau manifolds equipped with their Ricci-flat Kahler metrics, with an emphasis on the case when the metrics are volume collapsing.
The paper studies how spaces collapse to Alexandrov spaces with mild singularities.
Prove that collapsing CSC metrics can be perturbed to invariant collapsing CSC metrics.
In this paper, an n-dimensional complete open manifold with nonnegative Ricci curvature and collapsing volume has been investigated. If its radial sectional curvature bounded from below, it shows that such a manifold is of finite topological type under some restrictions shown below.
Durhuus and Jonsson (1995) introduced the class of "locally constructible" (LC) triangulated manifolds and showed that all the LC 2- and 3-manifolds are spheres. We show here that for each d>3 some LC d-manifolds are not spheres. We prove this result by studying how to collapse products of manifolds with exactly one fa…
MMCGAN uses explicit manifold learning to improve GAN performance.
Proves Euler characteristic of collapsing Alexandrov spaces.
Nonpositive towers property in 3-manifolds spines.
Paper shows stability of metric reconstruction for orbifolds from spectral data.
We give relationships between the vanishing of the A-hat genus and the possibility that a spin manifold can collapse with curvature bounded below.
Sphere theorems for specific manifolds with curvature constraints.
The paper explores embedding Ricci flow solutions in flag manifolds.
Unified proof of smooth fibration theorems for collapsed manifolds.
The study defines a canonical nilpotent structure for certain collapsed manifolds.
We consider the geometric inverse problem of determining a closed Riemannian manifold from measurements of the heat kernel in an open subset of the manifold. In this paper we analyze the stability of this problem in the class of -dimensional Riemannian manifolds with bounded diameter and sectional curvature. It is w…
The Kähler-Ricci flow on certain manifolds collapses to a canonical metric.
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 -manifold of bounded sectional curvature are -bi-Lipchitz equivalent and sufficient collapsed (depending on and ), then up to a diffeo…
Neural collapse occurs in normalized features over a Riemannian manifold.
Restrictions are obtained on the topology of a compact divergence-free null hypersurface in a four-dimensional Lorentzian manifold whose Ricci tensor is zero or satisfies some weaker conditions. This is done by showing that each null hypersurface of this type can be used to construct a family of three-dimensional Riema…