arXiv research

Ricci flow smooths locally collapsing manifolds with controlled curvature.

Enhances Ricci flow theorem with scalar curvature bound.

LDReg addresses local dimensional collapse in self-supervised learning.

New method controls posterior collapse in VAEs without network architecture constraints.

Self-attention networks localize when eigenspectrum variance is small.

We study the geometry and topology of Riemannian 3-orbifolds which are locally volume collapsed with respect to a curvature scale. We show that a sufficiently collapsed closed 3-orbifold without bad 2-suborbifolds either admits a metric of nonnegative sectional curvature or satisfies Thurston's Geometrization Conjectur…

We localize the entropy functionals of G. Perelman and generalize his no-local-collapsing theorem and pseudo-locality theorem. Our generalization is technically inspired by further development of Li-Yau estimate along the Ricci flow. It can be used to show the Gromov-Hausdorff convergence of the Kähler Ricci flow on ea…

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.

Generalizes tools for studying collapsed manifolds to new geometry.

Topology of non-orientable spaces without boundary is studied.

Posterior collapse in Variational Autoencoders (VAEs) arises when the variational posterior distribution closely matches the prior for a subset of latent variables. This paper presents a simple and intuitive explanation for posterior collapse through the analysis of linear VAEs and their direct correspondence with Prob…

Study shows local topologies of certain geometric spaces.

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 …

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…

Let $X$ be a non-collapsing Ricci limit space and let $x\in X$. We show that for any $ε>0$, there is $r>0$ such that every loop in $B_t(x)$ is contractible in $B_{(1+ε)t}(x)$, where $t\in(0,r]$. In particular, $X$ is semi-locally simply connected.

The study defines a canonical nilpotent structure for certain collapsed manifolds.

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…

Study collapsing geometry with Ricci curvature, proving Kähler metrics and Killing structures.

We construct the first explicit example of a simplicial 3-ball B_{15,66} that is not collapsible. It has only 15 vertices. We exhibit a second 3-ball B_{12,38} with 12 vertices that is collapsible and evasive, but not shellable. Finally, we present the first explicit triangulation of a 3-sphere S_{18, 125} (with only 1…

New collapsing mechanism for G2-manifolds discovered.

Gravitational instantons collapse to a punctured plane with a special Kahler metric.

New families of Ricci solitons found with collapsing volume.

The paper studies how spaces collapse to Alexandrov spaces with mild singularities.

The Degasperis-Procesi equation's solutions define pseudospherical metrics and can lead to surface collapse.

An orbifold $X$ is locally homeomorphic to $G_x\backslash B_r(0)$, where $G_x$ is a finite group acting on $B_r(0)\subset{\mathbb R}^n$, so that $G_x(0)=0$. For collapsing orbifolds with isolated singularities, we show there is a uniform bound in $|G_x|$.

The Kähler-Ricci flow on certain manifolds collapses to a canonical metric.

This article is a sequel to the book `Ricci Flow and the Poincare Conjecture' by the same authors. Using the main results of that book we establish the Geometrization Conjecture for all compact, orientable three-manifolds following the approach indicated by Perelman in his preprints on the subject. This approach is to …

Assuming a lower bound on the Ricci curvature of a complete Riemannian manifold, for $p< 1/2$ we show the existence of bounds on the local $L^p$ norm of the Ricci curvature that depend only on the dimension and which improve with volume collapse.

Uniform volume estimate for Kähler metrics in big cohomology classes.

Uniform estimates for Kaehler metrics' diameters and volumes.

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…

The paper studies a relative version of non-positive immersion for 2-complex pairs and shows conditions under which a transitivity law holds.

In this paper, we show that Generative Adversarial Networks (GANs) suffer from catastrophic forgetting even when they are trained to approximate a single target distribution. We show that GAN training is a continual learning problem in which the sequence of changing model distributions is the sequence of tasks to the d…

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…

The paper extends a theorem to manifolds with local Ricci bounded covering geometry.

We obtain new topological information about the local structure of collapsing under a lower sectional curvature bound. As an application we prove a new sphere theorem and obtain a partial result towards the conjecture that not every Alexandrov space can be obtained as a limit of a sequence of Riemannian manifolds with …

We analyze the topology and geometry of a polyhedron of dimension 2 according to the minimum size of a cover by PL collapsible polyhedra. We provide partial characterizations of the polyhedra of dimension 2 that can be decomposed as the union of two PL collapsible subpolyhedra in terms of their simple homotopy type and…

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 …

New analysis shows how attention masks and LayerNorm prevent rank collapse in transformers.

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 $L_0$-bi-Lipchitz equivalent and sufficient collapsed (depending on $L_0$ and $n$), then up to a diffeo…

Variational autoencoders (VAEs) hold great potential for modelling text, as they could in theory separate high-level semantic and syntactic properties from local regularities of natural language. Practically, however, VAEs with autoregressive decoders often suffer from posterior collapse, a phenomenon where the model l…

We prove a uniform C^alpha estimate for collapsing Calabi-Yau metrics on the total space of a proper holomorphic submersion over the unit ball in C^m. The usual methods of Calabi, Evans-Krylov, and Caffarelli do not apply to this setting because the background geometry degenerates. We instead rely on blowup arguments a…

Constructs initial data for multiple black holes with specified ADM parameters.

Paper shows stability of metric reconstruction for orbifolds from spectral data.

This paper examines how different loss functions affect neural network features and performance.

Study limits of curved spaces with boundaries.

We study the collapsing behavior of the Kaehler-Ricci flow on a compact Kaehler manifold X admitting a holomorphic submersion X -> S coming from its canonical class, where S is a Kaehler manifold with dim S < dim X. We show that the flow metric degenerates at exactly the rate of e^{-t} as predicted by the cohomology in…

Formula connects curvature to volume in special geometric spaces.