arXiv research

Users create melodies with AI, harmonized in the style of Bach.

Classifies doodles into prime and super prime types, describing them with doodle codes.

Alexander invariant created for doodles, vanishes on unlinked doodles.

Doodles were introduced in [R. Fenn and P. Taylor, Introducing doodles, Topology of low-dimensional manifolds, pp. 37--43, Lecture Notes in Math., 722, Springer, Berlin, 1979] but were restricted to embedded circles in the 2-sphere. Khovanov, [M. Khovanov, Doodle groups, Trans. Amer. Math. Soc. 349 (1997), 2297--2315],…

Doodles link to commutator identities in a 2-sphere.

Paper introduces skew-symmetric matrices for virtual doodle classification.

Complete invariant defined for doodles on a sphere.

We discuss Gauss codes of virtual diagrams and virtual doodles. The notion of a left canonical Gauss code is introduced and it is shown that oriented virtual doodles are uniquely presented by left canonical Gauss codes.

A virtual doodle is an equivalence class of virtual diagrams under an equivalence relation generated by flat version of classical Reidemesiter moves and virtual Reidemsiter moves such that Reidemeister moves of type 3 are forbidden. In this paper we discuss colorings of virtual diagrams using an algebra, called a doodl…

Cactus doodles are geometric objects derived from cactus groups.

Paper defines doodles on closed surfaces, unifying classical and virtual theories.

Paper shows how to represent Milnor's triple linking number using chord diagrams and doodle invariants.

Computes invariants distinguishing between immersions and embeddings of doodles and blobs on surfaces.

In 1997 M.~Khovanov proved that any doodle can be presented as closure of twin, this result is analogue of classical Alexander's theorem for braids and links. We give a description of twins that have equivalent closures, this theorem is analogue of classical Markov theorem.

DOODL learns shared spectral dynamics across related dynamical systems.

The paper introduces and characterizes almost ω-Bach solitons on various product manifolds.

Study of Bach flow on specific nilmanifolds, converging to a soliton.

The paper extends results on Bach-flat solitons to new types.

In this paper, we study vacuum static spaces with the complete divergence of the Bach tensor and Weyl tensor. First, we prove that the vanishing of complete divergence of the Bach tensor and Weyl tensor implies the harmonicity of the metric, and we present examples in which these conditions do not imply Bach flatness. …

Study on gradient ρ-Einstein solitons with radially nonnegative Bach tensor.

Researchers study solitons on homogeneous manifolds, proving properties of specific types of solitons.

The paper generalizes Bach and Einstein equations with a field.

New findings on shrinking Ricci solitons with vanishing Bach-like tensors.

We prove a removal of singularities result for Bach-flat metrics in dimension 4 under the assumption of bounded L^2 norm of curvature, bounded Sobolev constant and a volume growth bound. This result extends the removal of singularities result for special classes of Bach-flat metrics obtained in \cite{TVMOD}. For the pr…

Introduces conformal Bach flow and proves its well-posedness and backward uniqueness.

The purpose of this article is to investigate Bach-flat critical metrics of the volume functional on a compact manifold $M$ with boundary $\partial M.$ Here, we prove that a Bach-flat critical metric of the volume functional on a simply connected 4-dimensional manifold with boundary isometric to a standard sphere must …

Grading function evaluates Bach-style chorales, outperforming human experts.

We construct examples of Bach-flat gradient Ricci solitons which are neither half conformally flat nor conformally Einstein.

The paper proves stability for a modified Bach flow on various manifolds.

In this paper, we classify n-dimensional (n>3) complete Bach-flat gradient shrinking Ricci solitons. More precisely, we prove that any 4-dimensional Bach-flat gradient shrinking Ricci soliton is either Einstein, or locally conformally flat hence a finite quotient of the Gaussian shrinking soliton $R^4$ or the round cyl…

The paper proves rigidity of certain solitons with specific properties.

New metrics found with specific curvature properties on 4D manifolds.

Around 2007, A. Chang, J. Qing, and P. Yang proved a conformal gap theorem for Bach-flat metrics with round sphere as the model case. In this article, we extend this result to prove conformally invariant gap theorems for Bach-flat $4$-manifolds with $(\mathbb{CP}^2, g_{FS})$ and $(\mathbb{S}^2\times\mathbb{S}^2,g_{prod…

The goal of this article is to study the geometry of Bach-flat noncompact steady quasi-Einstein manifolds. We show that a Bach-flat noncompact steady quasi-Einstein manifold $(M^{n},\,g)$ with positive Ricci curvature such that its potential function has at least one critical point must be a warped product with Einstei…

We establish the existence of solvable Lie groups of dimension 4 and left-invariant Riemannian metrics with zero Bach tensor which are neither conformally Einstein nor half conformally flat.

In this paper we prove that any $n$-dimensional ($n\ge 4$) complete Bach-flat gradient steady Ricci soliton with positive Ricci curvature is isometric to the Bryant soliton. We also show that a three-dimensional gradient steady Ricci soliton with divergence-free Bach tensor is either flat or isometric to the Bryant sol…

The paper classifies 10 antipodal pairings of self-dual maps.

Electrostatic systems with specific tensors are locally conformally flat.

Researchers prove a new inequality for special Riemannian manifolds.

We prove that an n($\geq$ 4)-dimensional compact Bach-flat manifold with positive constant $σ_2$ is an Einstein manifold, provided that its Weyl curvature satisfies a suitable pinching condition.

Proves Alexander and Markov theorems for higher genus virtual doodles.

In this paper, we prove some rigidity theorems for compact Bach-flat $n$-manifold with the positive constant scalar curvature. In particular, our conditions in Theorem 1.4 have the additional properties of being sharp.

It has been observed by Maldacena that one can extract asymptotically anti-de Sitter Einstein $4$-metrics from Bach-flat spacetimes by imposing simple principles and data choices. We cast this problem in a conformally compact Riemannian setting. Following an approach pioneered by Fefferman and Graham for the Einstein e…

In this paper we show that a compact warped product Einstein manifold with vanishing Bach tensor of dimension $n \geq 4$ is a finite quotient of a warped product with $(n-1)$-dimensional Einstein fiber. The fiber has constant curvature if $n=4$.

Rigidity theorem for special metrics on 4-manifolds.

On a compact $n$-dimensional manifold $M$, it is well known that a critical metric of the total scalar curvature, restricted to the space of metrics with unit volume, is Einstein. It has been conjectured that a critical metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvat…

For complete Riemannian manifolds with vanishing Bach tensor and positive constant scalar curvature, we provide a rigidity theorem characterized by some pointwise inequalities. Furthermore, we prove some rigidity results under an inequality involving $L^{\frac{n}{2}}$-norm of the Weyl curvature, the traceless Ricci cur…

We use the modified Riemannian extension of an affine surface to construct Bach flat manifolds. As all these examples are VSI (vanishing scalar invariants), we shall construct scalar invariants which are not of Weyl type to distinguish them. We illustrate this phenomena in the context of homogeneous affine surfaces.