arXiv research

The paper proves scalar curvature decay for uniformly contractible manifolds with finite asymptotic dimension.

Whenever a finitely generated group $G$ acts properly discontinuously by isometries on a metric space $X$, there is an induced uniform embedding (a Lipschitz and uniformly proper map) $ρ: G \rightarrow X$ given by mapping $G$ to an orbit. We study when there is a difference between a finitely generated group $G$ acting…

We show that a space with a finite asymptotic dimension is embeddable in a non-positively curved manifold. Then we prove that if a uniformly contractible manifold X is uniformly embeddable in $\R^n$ or non-positively curved n-dimensional simply connected manifold then $X\times\R^n$ is integrally hyperspherical. If a un…

Uniformly extend maps on Hadamard manifolds with curvature constraints.

This paper examines limits of Riemannian 2-manifolds with bounded curvature.

The simplicial volume of non-R^3 contractible 3-manifolds is infinite.

No 5D aspherical manifolds can have uniformly positive scalar curvature.

Curvature conditions distinguish Euclidean space and disks in contractible manifolds.

Study finds topological restrictions on 4-manifolds with uniformly positive scalar curvature.

We consider compact convex hypersurfaces contracting by functions of their curvature. Under the mean curvature flow, uniformly convex smooth initial hypersurfaces evolve to remain smooth and uniformly convex, and contract to points after finite time. The same holds if the initial data is only weakly convex or non-smoot…

We show that an enlargeable Riemannian metric on a (possibly nonspin) manifold cannot have uniformly positive scalar curvature. This extends a well-known result of Gromov and Lawson to the nonspin setting. We also prove that every noncompact manifold admits a nonenlargeable metric. In proving the first result, we use t…

This paper proves properties of uniformly hyperbolic sets and constructs Markov partitions.

We prove that convex hypersurfaces in ${\mathbb R}^{n+1}$ contracting under the flow by any power $α>\frac{1}{n+2}$ of the Gauss curvature converge (after rescaling to fixed volume) to a limit which is a smooth, uniformly convex self-similar contracting solution of the flow. Under additional central symmetry of the ini…

We complete the proof of the Generalized Smale Conjecture, apart from the case of $RP^3$, and give a new proof of Gabai's theorem for hyperbolic 3-manifolds. We use an approach based on Ricci flow through singularities, which applies uniformly to spherical space forms other than $S^3$ and $RP^3$ and hyperbolic manifold…

We develop some basic Lipschitz homotopy technique and apply it to manifolds with finite asymptotic dimension. In particular we show that the Higson compactification of a uniformly contractible manifold is mod $p$ acyclic in the finite dimensional case. Then we give an alternative proof of the Higher Signature Novikov …

Sharp growth tightness proven for group quotients.

Study on singularities of solutions to Hamilton-Jacobi equations on manifolds.

Study of Brown--York mass for four-dimensional asymptotically flat manifolds.

The paper analyzes game theory in convertible contracts during liquidity events.

New algorithm uniformly samples high-dimensional convex bodies efficiently.

In the paper portfolio optimization over long run risk sensitive criterion is considered. It is assumed that economic factors which stimulate asset prices are ergodic but non necessarily uniformly ergodic. Solution to suitable Bellman equation using local span contraction with weighted norms is shown. The form of optim…

We construct a compact metric space that has any other compact metric space as a tangent, with respect to the Gromov-Hausdorff distance, at all points. Furthermore, we give examples of compact sets in the Euclidean unit cube, that have almost any other compact set of the cube as a tangent at all points or just in a den…

One can define what it means for a compact manifold with corners to be a "contractible manifold with contractible faces." Two combinatorially equivalent, contractible manifolds with contractible faces are diffeomorphic if and only if their 4-dimensional faces are diffeomorphic. It follows that two simple convex polytop…

The paper studies circle packings using renormalization and subdivision rules.

Proposes a new generalization bound for Bayesian deep nets without strict assumptions.

Compact Kähler manifolds with positive curvature have contractible covers.

Homotopy equivalence shown between complex and thickened versions of manifolds.

Theory of relatively Anosov representations using flow methods.

Classifies 3-manifolds with uniformly positive scalar curvature.

Establishes exponential contraction in Wasserstein distance on manifolds and flows.

Exotic diffeomorphism survives stabilizations on a contractible 4-manifold.

Study shows how discrete graph curvature relates to manifold curvature.

We prove that the restricted holonomy group of a complete smooth solution to the Ricci flow of uniformly bounded curvature cannot spontaneously contract in finite time; it follows, then, from an earlier result of Hamilton that the holonomy group is exactly preserved by the equation. In particular, a solution to the Ric…

Transformer pretraining yields strong EB performance without explicit adaptation.

Constructs uniformly positive scalar curvature metrics on open manifolds

We establish the equivalence between the family of closed uniformly regular Riemannian manifolds and the class of complete manifolds with bounded geometry.

Lickorish has constructed large families of contractible 4--manifolds that have knotted embeddings in the 4--sphere and has also shown that every finitely presented perfect group with balanced presentation occurs as the fundamental group of the complement of a knotted contractible manifold. Here we make a few observati…

The paper constructs contractible manifolds with knotted spheres.

Alexander trick applied to homology spheres for manifold homeomorphisms.

Non-linear Hopf manifolds can be embedded into linear ones and admit LCK metrics.

Examples are given to show that some compact contractible 4-manifolds can be knotted in the 4-sphere. It is then proved that any finitely presented perfect group with a balanced presentation is a knot group for an embedding of some contractible 4-manifold in the 4-sphere.

Uniformly positive scalar curvature implies a lower bound on injectivity radius.

Study of contractible manifolds and their twists to determine if they are $S^4$.

Formula found for skinning map contraction in hyperbolic geometry.

Develops Lefschetz theory for noncompact manifolds.

We prove that a contractible orbifold is a manifold.

Paper tackles which 3-spheres bound contractible 4-manifolds or homology 4-balls.

We show that on smooth minimal surfaces of general type, the Kähler-Ricci flow starting at any initial Kähler metric converges in the Gromov-Hausdorff sense to a Kähler-Einstein orbifold surface. In particular, the diameter of the evolving metrics is uniformly bounded for all time and the Kähler-Ricci flow contracts al…