Homogeneous Hypersurfaces Research Articles

Backreaction in Numerical Relativity: Averaging on Newtonian gauge-like hypersurfaces in Einstein Toolkit cosmological simulations

We introduce a spatial averaging scheme and use it to study the evolution of spatial averages in large-scale simulations of cosmological structure formation performed with the Einstein Toolkit. The averages are performed on the spatial hypersurfaces of the simulation setup which, at least initially, represent the hypersurfaces of statistical homogeneity and isotropy. We find only negligible cosmic backreaction on these hypersurfaces even on very small scales, but find significant curvature fluctuations of up to 10% in ΩR for sub-volumes with radius ∼200 Mpc and even larger fluctuations in smaller sub-volumes. In addition, we quantify fluid flow in and out of these sub-volumes. We find this to be significant, up to a 5% change in the density between redshift z=1 and z=0 of a single sphere of radius ∼200 Mpc (and larger for smaller spheres). We suggest this may be important for studies basing averages on volumes co-moving with the simulation hypersurfaces.

Three-dimensional quasi-umbilical locally strongly convex homogeneous affine hypersurfaces

The study of strongly convex, locally homogeneous hypersurfaces in the affine space ℝn+1 has a rich history. In dimension 2, these hypersurfaces were first examined in Guggenheimer's 1977 book “Differential Geometry” [4], and the classification was completed by Nomizu and Sasaki in their 1991 paper “A new model of unimodular-affinely homogeneous surfaces” [6]. Apart from the work of Magid and Vrancken in their 1995 paper [5] and Ooguri’s 2013 paper [9], most studies have focused on locally strongly convex affine hypersurfaces. These are characterized by an induced affine metric that is positive definite, and have been extensively studied in Sasaki’s 1980 paper “Hyperbolic affine hyperspheres” [10]. When the hypersurface is an affine sphere, where the shape operator has all eigenvalues equal to zero, Dillen and Vrancken provided a complete classification in a series of papers. In dimension 3, their 1993 paper “The classification of 3-dimensional locally strongly convex homogeneous affine hypersurfaces” [3] offers a comprehensive classification for any dimensions of eigenvalues of the shape operator. In dimension 4, the quasi-umbilical case was explored by Dillen and Vrancken in their 1994 paper “Quasi-umbilical, locally strongly convex homogeneous affine hypersurfaces” [2]. The case where the affine shape operator has two distinct eigenvalues, each with a multiplicity of two, was investigated by Chikh-Salah and Vrancken in their 2017 paper “Four-dimensional locally strongly convex homogeneous affine hypersurfaces” [1]. In this paper, we provide a different proof of the final theorem in Dillen and Vrancken’s [3], utilizing Mathematica software to calculate all Christoffel symbol functions.

Möbius homogeneous hypersurfaces in [formula omitted]

Let M(Sn+1) denote the Möbius transformation group of the (n+1)-dimensional sphere Sn+1. A hypersurface x:Mn→Sn+1 is called a Möbius homogeneous hypersurface if there exists a subgroup G of M(Sn+1) such that the orbit G⋅p=x(Mn),p∈x(Mn). In this paper, the Möbius homogeneous hypersurfaces are classified completely up to a Möbius transformation of Sn+1.

Classification of homogeneous hypersurfaces in some noncompact symmetric spaces of rank two

We classify, up to isometric congruence, the homogeneous hypersurfaces in the Riemannian symmetric spaces textrm{SL}(3,mathbb {H})/textrm{Sp}(3), textrm{SO}(5,mathbb {C})/textrm{SO}(5), and textrm{Gr}^*(2,mathbb {C}^{n+4}) = textrm{SU}(n+2,2)/textrm{S}(textrm{U}(n+2)textrm{U}(2)), , n geqslant 1.

Jet schemes of quasi-homogeneous hypersurfaces and motivic monodromy conjecture for isolated quasi-homogeneous hypersurface singularities

Let $k$ be a field. Let $m\in\mathbf{N}_{>0}$ be a positive integer. Let $f\in k[x_1,\ldots,x_m]$ be a polynomial with degree $d\geq 1$ and associated hypersurface $H:=H(f):=\mathrm{Spec}(k[x_1,\ldots,x_m]/\langle f\rangle)$. In this article, we firstly provide a structure property of the weighted-homogeneity of $f$ in terms of the jet schemes ${\mathscr{L}_{H}}$ of $H$. As a by-product, we deduce from this property a new and very \linebreak effective method for the computation of the motivic Poincaré power series $P_H(T):=\sum_{n\geq 0} [\mathscr{L}_{n}(H)]T^n\in K_0(\mathrm{Var}_k)[[T]]$ associated with a homogeneous hypersurface $H$ with a single isolated singularity at the origin $\frak o$ (and more generally with a specific class of isolated quasi-homogeneous hypersurface singularities). With this point of view we obtain various consequences. For the considered class of varieties, our method provides a characteristic-free proof of the rationality of $P_H(T)$ which does not use motivic integration nor the existence of resolutions of singularities; we obtain a precise description of the numerator and the possible poles in the rational expression of $P_H(T)$; when the field is assumed to be of characteristic zero, this allows us to prove the validity of the motivic monodromy conjecture.

Homogeneous Einstein metrics on Euclidean spaces are Einstein solvmanifolds

We show that homogeneous Einstein metrics on Euclidean spaces are Einstein solvmanifolds, using that they admit periodic, integrally minimal foliations by homogeneous hypersurfaces. For the geometric flow induced by the orbit-Einstein condition, we construct a Lyapunov function based on curvature estimates which come from real GIT.

Holomorphically homogeneous real hypersurfaces in $\mathbb {C}^3$

We give a complete description and classification of locally homogeneous real hypersurfaces in $\mathbb C^3$. Various groups of mathematicians have been studying this problem in the last 25 years, and several significant classes of hypersurfaces under consideration have been studied and classified. The final results in the classification problem presented in this paper are obtained by using the classification of abstract 5-dimensional real Lie algebras, and by studying their representations by algebras of holomorphic vector fields in complex 3-space. The complete list of pairwise inequivalent hypersurfaces that we obtain contains 47 types of homogeneous hypersurfaces; some of the types are 1- or 2-parametric families, and each of the others is single hypersurface or a finite family of hypersurface.

On the Problem of Describing Holomorphically Homogeneous Real Hypersurfaces of Four-Dimensional Complex Spaces

We discuss two fragments of a large problem that extends the author’s recently completed similar studies in the space $$\mathbb C^3$$ to the next dimension. One of the fragments is related to the local description of nonspherical holomorphically homogeneous strictly pseudoconvex hypersurfaces in $$\mathbb C^4$$ with stabilizers of submaximal dimension. Using the Moser normal form technique and the properties of subgroups of the unitary group $$\mathrm U(3)$$ , we show that up to holomorphic equivalence there exist only two such surfaces. Both of them are natural generalizations of known homogeneous hypersurfaces in the space $$\mathbb C^3$$ . In the second part of the paper, we consider a technique of holomorphic realization in $$\mathbb C^4$$ of abstract seven-dimensional Lie algebras that correspond, in particular, to homogeneous hypersurfaces with trivial stabilizer. Some sufficient conditions for the Lie algebras are obtained under which the orbits of all realizations of such algebras are Levi degenerate. The schemes of studying holomorphically homogeneous hypersurfaces that were used in the two-dimensional (E. Cartan) and three-dimensional (Doubrov, Medvedev, and The; Fels and Kaup; Beloshapka and Kossovskiy; Loboda) situations and resulted in full descriptions of such hypersurfaces turn out to be quite efficient in the case of greater dimension of the ambient space as well.

Time-Like Conformal Homogeneous Hypersurfaces with Three Distinct Principal Curvatures

Time-Like Conformal Homogeneous Hypersurfaces with Three Distinct Principal Curvatures

On the non-existence of negative weight derivations of the new moduli algebras of singularities

We propose a new conjecture about the non-existence of negative weight derivations of the new moduli algebras of weighted homogeneous hypersurface singularities and verify this conjecture up to dimension three.

Gauge theory and $\mathrm G_2$-geometry on Calabi–Yau links

The 7-dimensional link K of a weighted homogeneous hypersurface on the round 9-sphere in \mathbb{C}^5 has a nontrivial null Sasakian structure which is contact Calabi–Yau, in many cases. It admits a canonical co-calibrated \mathrm G_2 -structure \varphi induced by the Calabi–Yau 3-orbifold basic geometry. We distinguish these pairs (K,\varphi) by the Crowley–Nordström \mathbb{Z}_{48} -valued \nu invariant, for which we prove odd parity and provide an algorithmic formula. We describe moreover a natural Yang–Mills theory on such spaces, with many important features of the torsion-free case, such as a Chern–Simons formalism and topological energy bounds. In fact, compatible \mathrm G_2 -instantons on holomorphic Sasakian bundles over K are exactly the transversely Hermitian Yang–Mills connections. As a proof of principle, we obtain \mathrm G_2 -instantons over the Fermat quintic link from stable bundles over the smooth projective Fermat quintic, thus relating in a concrete example the Donaldson–Thomas theory of the quintic threefold with a conjectural \mathrm G_2 -instanton count.

Principal Curvatures of Homogeneous Hypersurfaces in a Grassmann Manifold $\widetilde{\text{Gr}}_{ 3}(\text{Im}\mathbb{O})$ by the $G_2$-action

We compute the principal curvatures of homogeneous hypersurfaces in a Grassmann manifold $\widetilde{\text{Gr}}_{ 3}(\text{Im}\mathbb{O})$ by the $G_2$-action. As applications, we show that there is a unique orbit which is an austere submanifold, and that there are just two orbits which are proper biharmonic homogeneous hypersurfaces. We also show that the austere orbit is a weakly reflective submanifold.

On orbits of action of 5-dimensional non-solvable Lie algebras in three-dimensional complex space

After the description by E. Cartan in 1932 holomorphically homogeneous real hypersurfaces of two-dimensional complex spaces, a similar study in the 3-dimensional case remains incomplete. In a series of works performed by several international teams of authors, the problem is reduced to describing homogeneous surfaces that are non-degenerate in Levi sense and have exactly 5-dimensional Lie algebras of holomorphic vector fields. In this paper, precisely such homogeneous surfaces are investigated. At the same time, a significant part of the extensive list of abstract 5-dimensional Lie algebras does not provide new examples of homogeneity. A complete description of the orbits of 5-dimensional non-solvable Lie algebras in a three-dimensional complex space, given in the paper, includes examples of new homogeneous hypersurfaces. The presented results bring to finish a large-scale scientific study of interest to various branches of mathematics.

Hypersurface-homogeneous modified holographic Ricci dark energy cosmological model by hybrid expansion law in Saez–Ballester theory of gravitation

The main motive of this investigation is to study the behavior of cosmological model in the presence of matter and a modified holographic Ricci dark energy for homogeneous hypersurface in the scalar tensor theory of gravitation, proposed by Saez–Ballester (Phys. Lett. A, 113, 467 (1986)). The hybrid expansion law (Akarsu et al., JCAP, 01, 022 (2014)) has been used to get a determinate solution. The physical condition that is shear scalar proportional to the expansion scalar is used to obtain the solution of the field equations. The various physical and geometrical aspects of the model are also discussed.

Stability of associated forms

We show that the associated form, or, equivalently, a Macaulay inverse system, of an Artinian complete intersection of type ( d , … , d ) (d,\dots , d) is polystable. As an application, we obtain an invariant-theoretic variant of the Mather-Yau theorem for homogeneous hypersurface singularities.

On Holomorphic Realizations of Nilpotent Lie Algebras

Realizations of five-dimensional Lie algebras as algebras of holomorphic vector fields on homogeneous real hypersurfaces of a three-dimensional complex space are studied. In view of already known results, in the problem of describing such varieties only Levy nondegenerate hypersurfaces with exactly five-dimensional Lie algebras are of interest. It is shown that only two of the nine existing distinct nilpotent Lie algebras admit realizations associated with such varieties, and the varieties corresponding to these exceptional algebras are standard quadrics.

Non-existence of negative weight derivations on positively graded Artinian algebras

Let R = C [ x 1 , x 2 , … , x n ] / ( f 1 , … , f m ) R= {\Bbb C}[x_1,x_2,\ldots , x_n]/(f_1,\ldots , f_m) be a positively graded Artinian algebra. A long-standing conjecture in algebraic geometry, differential geometry, and rational homotopy theory is the non-existence of negative weight derivations on R R . Alexsandrov conjectured that there are no negative weight derivations when R R is a complete intersection algebra, and Yau conjectured there are no negative weight derivations on R R when R R is the moduli algebra of a weighted homogeneous hypersurface singularity. This problem is also important in rational homotopy theory and differential geometry. In this paper we prove the non-existence of negative weight derivations on R R when the degrees of f 1 , … , f m f_1,\ldots ,f_m are bounded below by a constant C C depending only on the weights of x 1 , … , x n x_1,\ldots ,x_n . Moreover this bound C C is improved in several special cases.

Spacelike Möbius Hypersurfaces in Four Dimensional Lorentzian Space Form

In this paper, we first set up an alternative fundamental theory of Mobius geometry for any umbilic-free spacelike hypersurfaces in four dimensional Lorentzian space form, and prove the hypersurfaces can be determined completely by a system consisting of a function W and a tangent frame {Ei}. Then we give a complete classification for spacelike Mobius homogeneous hypersurfaces in four dimensional Lorentzian space form. They are either Mobius equivalent to spacelike Dupin hypersurfaces or to some cylinders constructed from logarithmic curves and hyperbolic logarithmic spirals. Some of them have parallel para-Blaschke tensors with non-vanishing Mobius form.

Computation methods of logarithmic vector fields associated to semi-weighted homogeneous isolated hypersurface singularities

Methods for computing logarithmic vector fields along a semi-weighted homogeneous hypersurface with an isolated singularity are considered in the context of symbolic computation. The main idea of our approach is based on the concept of polar variety and of algebraic local cohomology. New algorithms are introduced for computing a set of generators of the modules of logarithmic vector fields. The keys of the resulting algorithms are a notion of parametric syzygy system and that of parametric local cohomology system.

Characterizations of three homogeneous real hypersurfaces in a complex projective space

In an $n$-dimensional complex hyperbolic space $\mathbb{C}H^n(c)$ of constant holomorphic sectional curvature $c (\lt 0)$, the horosphere HS, which is defined by ${\rm HS} = \lim_{r\to\infty}G(r)$, is one of nice examples in the class of real hypersurfaces. Here, $G(r)$ is a geodesic sphere of radius $r$ $(0 \lt r \lt \infty)$ in $\mathbb{C}H^n(c)$. The second author ([14]) gave a geometric characterization of HS. In this paper, motivated by this result, we study real hypersurfaces $M^{2n-1}$ isometrically immersed into an $n$-dimensional complex projective space $\mathbb{C}P^n(c)$ of constant holomorphic sectional curvature $c(\gt 0)$.