arXiv research

Asymptotic dimension of planes and graphs is at most three.

Unified curvature flows on 2D and 3D triangulated manifolds.

The article studies curvature operator behavior in 3D under Ricci flow.

Smoothness of Sklyanin algebras examined in 3D and 4D cases.

We derive a priori interior Hessian estimates for the special Lagrangian equation $σ_{2}=1$ in dimension three.

The goal of this article is to survey recent developments in the theory of contact structures in dimension three.

We prove the existence of a geometric characteristic submanifold for non-positively curved manifolds of any dimension greater than or equal to three. In dimension three, our result is a geometric version of the topological characteristic submanifold theorem due to Jaco, Shalen and Johannson.

The main contribution of our paper is to give a partial classification of the quasi-exactly solvable Lie algebras of first order differential operators in three variables, and to show how this can be applied to the construction of new quasi-exactly solvable Schrödinger operators in three dimensions.

Study on defect topology in smectic liquid crystals.

Bayesian method classifies shapes in 2D and 3D.

In this short note we prove that, in dimension three, flat metrics are the only complete metrics with non-negative scalar curvature which are critical for the $σ_{2}$-curvature functional.

Using an analogue of Myers' theorem for minimal surfaces and three dimensional topology, we prove the diameter sphere theorem for Ricci curvature in dimension three and a corresponding eigenvalue pinching theorem. This settles these two open problems for closed 3 manifolds with positive Ricci curvature since they are b…

Characterizes chains in 3D CR and para-CR structures.

$\mathcal{I}$-non-degenerate spaces are spacetimes that can be characterized uniquely by their scalar curvature invariants. The ultimate goal of the current work is to construct a basis for the scalar polynomial curvature invariants in three dimensional Lorentzian spacetimes. In particular, we seek a minimal set of alg…

Ricci flow connects 3D Ricci limit spaces to smooth manifolds.

In a previous paper, we proved a number of optimal rigidity results for Riemannian manifolds of dimension greater than four whose curvature satisfy an integral pinching. In this article, we use the same integral Bochner technique to extend the results in dimension three. Then, by using the classification of closed thre…

Variationality of conformal geodesics fails in higher dimensions.

Paper proves existence of solutions for complex surface diffusion equation.

Reservoir computer dimensions estimated using three methods.

Extremal length is a conformal invariant that transfers naturally to the discrete setting, giving square tilings as a natural combinatorial analog of conformal mappings. Recent work by S. Hersonsky has explored generalizing these ideas to three-dimensional cube tilings. The connections between discrete extremal length …

We prove a Goldberg-Sachs theorem in dimension three. To be precise, given a three-dimensional Lorentzian manifold satisfying the topological massive gravity equations, we provide necessary and sufficient conditions on the tracefree Ricci tensor for the existence of a null line distribution whose orthogonal complement …

Study properties of group rings of three-manifold groups.

We prove that a sequence of solutions of the Seiberg-Witten equation with multiple spinors in dimension three can degenerate only by converging (after rescaling) to a Fueter section of a bundle of moduli spaces of ASD instantons.

Abstract: Proves compactness theorem for generalized Seiberg-Witten equations in 3D.

We prove a modified version of Turbiner's conjecture in three dimensions and we give a counter-example to the original conjecture. The Lie algebraic Schrödinger operators corresponding to flat metrics of a certain restricted type are shown to separate partially in either Cartesian, cylindrical or spherical coordinates.

Develops methods for constructing exact, non-stationary solutions to Euler equations.

Compact support steady flow in 3D Euler equations found.

We describe the structure of the Ricci tensor on a locally homogeneous Lorentzian gradient Ricci soliton. In the non-steady case, we show the soliton is rigid in dimensions three and four. In the steady case, we give a complete classification in dimension three.

The approximability of a convex body is a number which measures the difficulty to approximate that body by polytopes. We prove that twice the approximability is equal to the volume entropy for a Hilbert geometry in dimension two end three and that in higher dimension it is a lower bound of the entropy. As a corollary w…

Research confirms a conjecture about complex manifolds with total Betti number three.

Finite subdivision rules in high dimensions are shown to be equivalent to 3D rules.

We formulate a notion of (uniform) asymptotic involutivity and show that it implies (unique) integrability of corank-1 continuous distributions in dimensions three or less. This generalizes and extends a classical theorem of Frobenius Theorem which says that an involutive C^1 distribution is uniquely integrable.

Study compares three performance metrics of Bangladeshi banks.

Quantifies scalar curvature under $C^0$ convergence, proving a refined version in all dimensions.

Criterion for solvability of complex 2-Hessian equation on compact Kähler manifolds.

Study classifies static potentials on 3-manifolds, proving one-dimensionality under specific conditions.

Expanding on subgroup dimension, this paper covers new dimensions and compactness.

A result is given to find points where a real valued function on the plane is not smooth. Provided this function is induced by a smooth mapping from three dimensions to the plane, from a function on surfaces in three dimensions. This has applications to numerical methods such as image processing.

We study three-dimensional Alexandrov spaces with a lower curvature bound, focusing on extending three classical results on three-dimensional manifolds: First, we show that a closed three-dimensional Alexandrov space of positive curvature, with at least one topological singularity, must be homeomorphic to the suspensio…

Study circle actions with exactly three fixed points on specific manifolds.

We show that every volume preserving codimension one Anosov flow on a closed Riemannian manifold of dimension greater than three admits a global cross section and is therefore topologically conjugate to a suspension of a linear toral automorphism. This proves a conjecture of Verjovsky from the 1970's in the volume pres…

We show that for generic Riemannian metrics on a closed spin manifold of dimension three the Dirac operator has only simple eigenvalues.

We develop tools to study the topology and geometry of self-affine fractals in dimension three and higher. We use the self-affine structure and obtain rather detailed information about the connectedness of interior and boundary sets, and on the dimensions and intersections of boundary sets. As an application, we descri…

Proves rigidity for eigenvalue estimate on three-manifolds.

We provide infinitely many examples of pairs of diffeomorphic, non simply connected K\" ahler manifolds of complex dimension three with different Kodaira dimensions. Also, in any possible Kodaira dimension we find infinitely many pairs of non deformation equivalent, diffeomorphic K\" ahler threefolds.

Sharp curvature estimates for expanding Ricci solitons in various dimensions.

New complex orthogonal structures found on a 3D manifold.

The notion of a braid is generalized into two and three dimensions. Two-dimensional braids are described by braid monodromies or graphics called charts. In this paper we introduce the notion of curtains, and show that three-dimensional braids are described by braid monodromies or curtains.