arXiv research

3-manifold volume simplified to one-fifth of traditional measure.

Cubic fourfolds have K-stability and admit Kähler-Einstein metrics.

We prove that any complete metric on R^3 minus a ball with non-negative Ricci curvature and quadratic Ricci-curvature decay, has cubic volume growth.

Proves K-stability of cubic threefolds and calculates their Kähler-Einstein metrics.

Asymptotically flat manifolds with Euclidean volume growth are known to be ALE. In this paper, we consider a class of asymptotically flat manifolds with slower volume growth and prove that their asymptotic geometry is that of a fibration over an ALE manifold. In particular, we show that gravitational instantons with cu…

The study connects cubic differentials to convex RP^2-structures and their ends.

Study finds the shortest triply periodic graph spanning a cubic lattice.

We present a conjecture, based on computational results, on the area minimizing way to enclose and separate two arbitrary volumes in the flat cubic 3-torus. For comparable small volumes, we prove that an area minimizing double bubble in the 3-torus is the standard double bubble from R^3.

The study proves that certain minimal hypersurfaces in 4D space must be planes.

This is the second part of the investigation started in [Stationary solutions and asymptotic flatness I]. We prove here that Strongly Stationary ends having cubic volume growth are Weakly Asymptotically Flat. Combined with the results of the previous paper this shows that Strongly Stationary ends are Asymptotically Fla…

The study shows properties of stable anisotropic minimal hypersurfaces in 4D space.

This paper provides a classification result for gravitational instantons with cubic volume growth and cyclic fundamental group at infinity. It proves that a complete hyperkähler manifold asymptotic to a circle fibration over the Euclidean three-space is either the standard $\rl^3 \times \sph^1$ or a multi-Taub-NUT mani…

In the space $\mathbb U^4$ of cubic forms of surfaces, regarded as a $G$-space and endowed with a natural invariant metric, the ratio of the volumes of those representing umbilic points with negative to those with positive indexes is evaluated in terms of the asymmetry of the metric, defined here. A connection of this …

We derive a recursion relation for hyperbolic string vertices and apply it to string field theory.

{\em Riemannian cubics} are curves in a manifold $M$ that satisfy a variational condition appropriate for interpolation problems. When $M$ is the rotation group SO(3), Riemannian cubics are track-summands of {\em Riemannian cubic splines}, used for motion planning of rigid bodies. Partial integrability results are know…

A 3D catenoid in 4D space is a minimal hypersurface that cannot be extended to a higher-dimensional half-space.

Computes the monodromy of cubic surfaces branching over smooth cubic curves.

Study of sub-Riemannian cubics in SU(2) for long-term dynamics.

Study shows minimum $δ$-invariant for smooth cubic surfaces.

The cubic lattice stick index of a knot type is the least number of sticks necessary to construct the knot type in the 3-dimensional cubic lattice. We present the cubic lattice stick index of various knots and links, including all (p,p+1)-torus knots, and show how composing and taking satellites can be used to obtain t…

This paper refines homotopy theory for cubical sets and uniform spaces.

According to our previous results, the conjugacy class of the involution induced by the complex conjugation in the homology of a real non-singular cubic fourfold determines the fourfold up to projective equivalence and deformation. Here, we show how to eliminate the projective equivalence and to obtain a pure deformati…

The paper studies conformally flat cubic metrics with isotropic curvature, finding they must be Minkowski.

The study shows that certain cubical presentations lead to aspherical spaces.

This paper completes the classification of cubic tessellations on 3-manifolds.

New cubic forms linked to η-invariants and mod 2 indices.

It is shown that there exist non-singular cubic surfaces in CP^3 containing 5 twistor lines. This is the maximum number of twistor fibres that a non-singular cubic can contain. Cubic surfaces in CP^3 with 5 twistor lines are classified up to transformations preserving the conformal structure of S^4.

Develops a new higher-order calculus using cubic algebra.

Solves infinite family of cubic polynomial problems.

We study global log canonical thresholds of cubic surfaces with canonical singularities, and we prove the existence of a Kahler-Einstein metric on two singular cubic surfaces.

The flow and Yamada polynomials of cubic graphs are studied, extending known identities and conjectures.

Computes cohomology of smooth cubic surfaces using simplicial resolution.

Study de Rham theory for cubical manifolds and quandles.

Motivated by applications in computational anatomy, we consider a second-order problem in the calculus of variations on object manifolds that are acted upon by Lie groups of smooth invertible transformations. This problem leads to solution curves known as Riemannian cubics on object manifolds that are endowed with norm…

Complete classification of centroaffine hypersurfaces with parallel cubic form.

A natural family of affine cubic surfaces arises from SL(2)-characters of the 4-holed sphere and the 1-holed torus. The ideal locus is a tritangent plane which is generic in the sense that the cubic curve at infinity consists of three lines pairwise intersecting in three double points. We show that every affine cubic s…

Study on cubic curves and their tangents, finding Zariski pairs.

Let N_0 = C^2/H be an isolated quotient singularity with H in U (2) a finite subgroup. We show that for any Q-Gorenstein smoothings of N_0 a nearby fiber admits ALE Ricci-flat Kahler metrics in any Kahler class. Moreover, we generalize Kronheimer's results on hyperkahler 4-manifolds, by giving an explicit classificatio…

Classifies hexagonal circular 3-webs with cubic polar curves.

Approximating complex curves with simple parametric curves is widely used in CAGD, CG, and CNC. This paper presents an algorithm to compute a certified approximation to a given parametric space curve with cubic B-spline curves. By certified, we mean that the approximation can approximate the given curve to any given pr…

In finance, one usually deals not with prices but with growth rates $R$, defined as the difference in logarithm between two consecutive prices. Here we consider not the trading volume, but rather the volume growth rate $\tilde R$, the difference in logarithm between two consecutive values of trading volume. To this end…

We show that, for Finsler spaces with cubic metric, Landsberg spaces are Berwaldian. Also, for decomposable metrics, we determine specific conditions for a space with cubic metric to be of Berwald type, thus refining the result in [6].

We give a generalization of the well-known result of E. Cartan on isoparametric cubics by showing that a homogeneous cubic polynomial solution of the eiconal equation $|\nabla f|^2=9|x|^4$ must be rotationally equivalent to either $x_n^3-3x_n(x_1^2+...+x_{n-1}^2)$, or to one of four exceptional Cartan cubic polynomials…

Researchers classify and visualize 5-cube cubical surfaces.

Study on choosing points on cubic curves, answering some questions about their flexibility.

No nontrivial automorphisms for cubic surfaces moduli space.

Study of algebraic dynamics on Markov cubics in tropical geometry.

Origami solves real cubic equations, revealing a specific curve.