Real Vector Fields Research Articles

Rotating Killing horizons in generic F(R) gravity theories

We discuss various properties of rotating Killing horizons in generic $F(R)$ theories of gravity in dimension four for spacetimes endowed with two commuting Killing vector fields. Assuming there is no curvature singularity anywhere on or outside the horizon, we construct a suitable $(3+1)$-foliation. We show that similar to Einstein's gravity, we must have $T_{ab}k^ak^b=0$ on the Killing horizon, where $k^a$ is a null geodesic tangent to the horizon. For axisymmetric spacetimes, the effective gravitational coupling $\sim\,F'^{-1}(R)$ should usually depend upon the polar coordinate and hence need not necessarily be a constant on the Killing horizon. We prove that the surface gravity of such a Killing horizon must be a constant, irrespective of whether $F'(R)$ is a constant there or not. We next apply these results to investigate some further basic features. In particular, we show that any hairy solution for the real massive vector field in such theories is clearly ruled out, as long as the potential of the scalar field generated in the corresponding Einstein's frame is a positive definite quantity.

Phase structure of a Yukawa-like model in the presence of magnetic background and boundaries

In this paper, we investigate the thermodynamics of a Yukawa-like model, constituted of a complex scalar field interacting with real scalar and vector fields, in the presence of an external magnetic field and boundaries. By making use of mean-field approximation, we analyze the phase structure of this model at effective chemical equilibrium, under change of values of the relevant parameters of the model, focusing on the influence of the boundaries on the phase structure. The findings reveal a strong dependence of the nature of the phase structure on temperature, magnetic background and size of compactified coordinate, with possibility of a two-step phase transition.

Phase structure of the scalar Yukawa model with compactified spatial dimensions

In this work, we investigate the thermodynamic behavior of the generalized scalar Yukawa model, composed of a complex scalar field interacting with real scalar and vector fields. In particular, boundary effects on the phase structure are discussed using methods of quantum field theory on toroidal topologies. We concentrate on the dependence of the thermodynamics with the number of compactified spatial dimensions. In this sense, the phase transitions are analyzed and compared with the system in the situations of one, two and three compactified spatial dimensions. Our findings suggest that the presence of more boundaries tends to inhibit the broken phase.

Analytic Hypoellipticity for a New Class of Sums of Squares of Vector Fields in $${\mathcal {R}}^3$$ R 3

This paper is devoted to a substantial generalization of previous work on the analytic hypoellipticity of sums of squares \(P=\sum _1^4X^2_j\) of real vector fields with real analytic coefficient in three variables. For p(x, y) quasi-homogeneous in (x, y), consider the vector fields $$\begin{aligned} X_1 = \frac{\partial }{\partial x}, \quad X_2=-\frac{\partial }{\partial y} + p(x,y)\frac{\partial }{\partial t}, \quad X_3=x^{n_1}\frac{\partial }{\partial t}, \quad X_4=y^{n_2}\frac{\partial }{\partial t}, \end{aligned}$$ \( n_1, n_2 \ne 0\). We show that the operator $$\begin{aligned} P=\sum _1^4 X_j^2, \end{aligned}$$ well known to be \(C^\infty \)-hypoelliptic, is actually analytic hypoelliptic near the origin in \({\mathcal {R}}^3\).

Regularity in Orlicz spaces for non-divergence degenerate elliptic equations on homogeneous groups

Let G be a homogeneous group, and let X1, X2, · · ·, Xp0 be left-invariant real vector fields on G that are homogeneous of degree one with respect to the dilation group of G and satisfy Hormander’s condition. We establish a regularity result in the Orlicz spaces for the following equation: $$Lu\left( x \right) = \sum\limits_{j = 1}^{{p_0}} {{a_{ij}}\left( x \right){X_i}{X_J}u\left( x \right)} = f\left( x \right)$$ , where aij(x) are real valued, bounded measurable functions defined on G, satisfying the uniform ellipticity condition, and belonging to the space VMO(G) with respect to the subelliptic metric induced by the vector fields X1, X2 · · ·, Xp0.

Real-Formal Orbital Rigidity for Germs of Real Analytic Vector Fields on the Real Plane

For nź2$n \geqslant 2$, we consider Vnź$\mathcal {V}^{\mathbb {R}}_{n}$ the class of germs of real analytic vector fields on ź2,0ź$\left (\mathbb {R}^{2}, \widehat {0}\right )$ with zero (nź1)-jet and nonzero n-jet. We prove, for generic germs of Vnź$\mathcal {V}^{\mathbb {R}}_{n}$, that the real-formal orbital equivalence implies the real-analytic orbital equivalence, that is, the real-formal orbital rigidity takes place. This is the real analytic version of Voronin's formal orbital rigidity theorem.

Cubic Differential Systems with Invariant Straight Lines of Total Multiplicity Eight possessing One Infinite Singularity

In this work we consider the problem of classifying all configurations of invariant lines of total multiplicity eight (including the line at infinity) of real planar cubic differential systems. The classification was initiated in Bujac and Vulpe (J Math Anal Appl 423:1025–1080, 2015) where the cubic systems which possess 4 distinct infinite singularities are considered and 17 such distinct configurations were detected. That work was continued in Bujac and Vulpe (Qual Theory Dyn Syst 14(1):109–137, 2015) where the classification was done for the family of cubic systems with three distinct infinite singularities and where the existence of 5 distinct configurations was proved. The cubic systems with 2 infinite singularities are considered in Bujac (Bul Acad Stiinte Repub Mold Mat 1:48–86, 2015) and Bujac and Vulpe (Classification of a subfamily of cubic differential systems with invariant straight lines of total multiplicity eight. Universitat Autonoma de Barselona, 35, 2015) where 25 configurations were constructed. This article deals with the last remaining subfamily of cubic systems. We prove here a classification theorem (Main Theorem) of real planar cubic vector fields which possess exactly one infinite singular point and eight invariant straight lines, including the line at infinity and including their multiplicities. We show that these systems could possess only 4 such configurations. We underline that the classifications of all the above mentioned families of cubic systems are taken modulo the action of the group of real affine transformations and time rescaling and are given in terms of invariant polynomials. The algebraic invariants and comitants allow one to verify for any given real cubic system whether or not it has invariant lines of total multiplicity eight, and to specify its configuration of lines endowed with their corresponding real singularities of this system. The calculations can be implemented on computer and the results can therefore be applied for any family of cubic systems, given in any normal form.

Eigenvalues of vector fields, Bott's residue formula and integral invariants

Given a compatible vector field on a compact connected almost-complex manifold, we show in this article that the multiplicities of eigenvalues among the zero point set of this vector field have intimate relations. We highlight a special case of our result and reinterpret it as a vanishing-type result in the framework of the celebrated Atiyah–Bott–Singer localization formula. This new point of view, via the Chern–Weil theory and a strengthened version of Bott's residue formula, can lead to an obstruction to Killing real holomorphic vector fields on compact Hermitian manifolds in terms of a curvature integral.

Infinitesimal Hartman-Grobman Theorem in Dimension Three.

In this paper we give the main ideas to show that a real analytic vector field in R3 with a singular point at the origin is locally topologically equivalent to its principal part defined through Newton polyhedra under non-degeneracy conditions.

Lower bounds of Dirichlet eigenvalues for some degenerate elliptic operators

Let \(\Omega \) be a bounded open domain in \(\mathbb {R}^n\) with smooth boundary and \(X=(X_1, X_2, \ldots , X_m)\) be a system of real smooth vector fields defined on \(\Omega \) with the boundary \(\partial \Omega \) which is non-characteristic for X. If X satisfies the Hormander’s condition, then the vector fields is finite degenerate and the sum of square operator \(\triangle _{X}=\sum _{j=1}^{m}X_j^2\) is a finitely degenerate elliptic operator, otherwise the operator \(-\triangle _{X}\) is called infinitely degenerate. If \(\lambda _j\) is the jth Dirichlet eigenvalue for \(-\triangle _{X}\) on \(\Omega \), then this paper shall study the lower bound estimates for \(\lambda _j\). Firstly, by using the sub-elliptic estimate directly, we shall give a simple lower bound estimates of \(\lambda _j\) for general finitely degenerate \(\triangle _{X}\) which is polynomial increasing in j. Secondly, if \(\triangle _{X}\) is so-called Grushin type degenerate elliptic operator, then we can give a precise lower bound estimates for \(\lambda _j\). Finally, by using logarithmic regularity estimate, for infinitely degenerate elliptic operator \(\triangle _{X}\) we prove that the lower bound estimates of \(\lambda _j\) will be logarithmic increasing in j.

Interior [formula omitted]-regularity for parabolic divergence equations of Hörmander’s type

Let X1,X2,…,Xq be a system of real smooth vector fields satisfying Hörmander’s condition in a bounded domain Ω⊂Rn. We consider the following parabolic equationsut+Xi∗(aij(x,t)Xju)=Xi∗fiinΩT, where Xi∗ is the formal adjoint of Xi, the coefficients aij(x,t) are real valued measurable functions defined in ΩT=Ω×(0,T], satisfying the uniform parabolic condition(0.1)μ|ξ|2≤∑i,j=1qaij(x,t)ξiξj≤μ−1|ξ|2, for almost every (x,t)∈Rn×R, every ξ∈Rq and some constant μ. We prove the interior W∗1,p(x,t)-regularity for weak solutions to the parabolic equations, under the assumptions that p(x,t) satisfies the strong log-Hölder continuity condition and the coefficients aij(x,t) belong to the space VMOloc∩L∞. Our method relies on a Gagliardo–Nirenberg inequality constituted on Hörmander’s vector fields and a certain Vitali covering lemma.

Weighted Sobolev–Morrey estimates for hypoelliptic operators with drift on homogeneous groups

Let X0,X1,…,Xq be left invariant real vector fields on the homogeneous group G and assume that X1,…,Xq are homogeneous of degree one and X0 is homogeneous of degree two. In this paper we consider the following hypoelliptic operator with driftL=∑i,j=1qaijXiXj+a0X0 where (aij) is a constant matrix satisfying the elliptic condition in Rq and a0≠0, and obtain weighted Sobolev–Morrey estimates and Morrey estimates with two weights of L by establishing boundedness of several singular integrals and interpolation lemmas on G.

Kähler manifolds with real holomorphic vector fields

For a Kahler manifold endowed with a weighted measure \(e^{-f}\,dv,\) the associated weighted Hodge Laplacian \(\Delta _{f}\) maps the space of \((p,q)\)-forms to itself if and only if the \((1,0)\)-part of the gradient vector field \(\nabla f\) is holomorphic. We use this fact to prove that for such \(f\), a finite energy \(f\)-harmonic function must be pluriharmonic. Motivated by this result, we verify that the same also holds true for \(f\)-harmonic maps into a strongly negatively curved manifold. Furthermore, we demonstrate that such \(f\)-harmonic maps must be constant if \(f\) has an isolated minimum point. In particular, this implies that for a compact Kahler manifold admitting such a function, there is no nontrivial homomorphism from its first fundamental group into that of a strongly negatively curved manifold.

Multivariable Newton-Puiseux Theorem for Generalised Quasianalytic Classes

We show how to solve explicitly an equation satisfied by a real function belonging to certain general quasianalytic classes. Examples of the classes under consideration are the collection of convergent generalised power series, a class of functions which contains some Dulac Transition Maps of real analytic planar vector fields, quasianalytic Denjoy-Carleman classes and the collection of multisummable series.

Cubic Systems with Invariant Straight Lines of Total Multiplicity Eight and with Three Distinct Infinite Singularities

In this article we prove a classification theorem (Main Theorem) of real planar cubic vector fields which possess eight invariant straight lines, including the line at infinity and including their multiplicities and in addition they possess three distinct infinite singularities. This classification, which is taken modulo the action of the group of real affine transformations and time rescaling, is given in terms of affine invariant polynomials. The invariant polynomials allow one to verify for any given real cubic system whether or not it has invariant straight lines of total multiplicity eight, and to specify its configuration of straight lines endowed with their corresponding real singularities of this system. The calculations can be implemented on computer and the results can therefore be applied for any family of cubic systems in this class, given in any normal form.

Cubic differential systems with invariant straight lines of total multiplicity eight and four distinct infinite singularities

In this article we prove a classification theorem (Main Theorem) of real planar cubic vector fields which possess four distinct infinite singularities and eight invariant straight lines, including the line at infinity and including their multiplicities. This classification, which is taken modulo the action of the group of real affine transformations and time rescaling, is given in terms of invariant polynomials. The algebraic invariants and comitants allow one to verify for any given real cubic system with four infinite distinct singularities whether or not it has invariant lines of total multiplicity eight, and to specify its configuration of lines endowed with their corresponding real singularities of this system. The calculations can be implemented on computer.

Local momentum space and the vector field

The local momentum space expansion for the real vector field is considered. Using Riemann normal coordinates we obtain an expansion of the Feynman Green function up and including terms that are quadratic in the curvature. The results are valid for a non-minimal operator such as that arising from a general Feynman type gauge fixing condition. The result is used to derive the first three terms in the asymptotic expansion for the coincidence limit of the heat kernel without taking the trace, thus obtaining the untraced heat kernel coefficients. The spacetime dimension is kept general before specializing to four dimensions for comparison with previously known results. As a further application we re-examine the anomalous trace of the stress-energy-momentum tensor for the Maxwell field and comment on the gauge dependence.

Briot-Bouquet’s Theorem in high dimension

Let X be a germ of holomorphic vector field at 0 2 Cn and let E be a linear subspace of Cn which is invariant for the linear part of X at 0. We give a suficient condition that imply the existence of a non-singular invariant manifold tangent to E at 0. It generalizes to higher dimensions the conditions in the classical Briot{Bouquet's Theorem: roughly speaking, we impose that the convex hull of the eigenvalues ui corresponding to E does not contain 0 and there are no resonances between the ui and the complementary eigenvalues. As an application, we propose an elementary proof of the analyticity of the local stable and unstable manifolds of a real analytic vector field at a singular point.

Index of singularities of real vector fields on singular hypersurfaces

Gomez-Mont, Seade and Verjovsky introduced an index, now called GSV-index, generalizing the Poincare-Hopf index to complex vector fields tangent to singular hypersurfaces. The GSV-index extends to the real case. This is a survey paper on the joint research with Gomez-Mont and Giraldo about calculating the GSV-index $\Ind_{V_\pm,0}(X)$ of a real vector field $X$ tangent to a singular hypersurface $V=f^{-1}(0)$. The index $\Ind_{V_{\pm,0}}(X)$ is calculated as a combination of several terms. Each term is given as a signature of some bilinear form on a local algebra associated to $f$ and $X$. Main ingredients in the proof are Gomez-Mont's formula for calculating the GSV-index on \emph{ singular complex} hypersurfaces and the formula of Eisenbud, Levine and Khimshiashvili for calculating the Poincare-Hopf index of a singularity of a \emph{real} vector field in $\R^{n+1}$

Lpand Schauder estimates for nonvariational operators structured on Hörmander vector fields with drift

We consider linear second order nonvariational partial differential operators of the kind a_{ij}X_{i}X_{j}+X_{0}, on a bounded domain of R^{n}, where the X_{i}'s (i=0,1,2,...,q, n>q+1) are real smooth vector fields satisfying H\"ormander's condition and a_{ij} (i,j=1,2,...,q) are real valued, bounded measurable functions, such that the matrix {a_{ij}} is symmetric and uniformly positive. We prove that if the coefficients a_{ij} are H\"older continuous with respect to the distance induced by the vector fields, then local Schauder estimates on X_{i}X_{j}u, X_{0}u hold; if the coefficients belong to the space VMO with respect to the distance induced by the vector fields, then local L^{p} estimates on X_{i}_{j}u, X_{0}u hold. The main novelty of the result is the presence of the drift term X_{0}, so that our class of operators covers, for instance, Kolmogorov-Fokker-Planck operators.