Invariant Hypersurfaces Research Articles

ON ALMOST r-PARACONTACT RIEMANNIAN MANIFOLD WITH A CERTAIN CONNECTION

In a Riemannian manifold, the existence of a new connection is proved. In particular cases, this connection reduces to several symmetric, semi-symmetric and quarter symmetric connections, even some of them are not introduced so far. So, in this paper, we define a quarter symmetric semi-metric connection in an almost r-paracontact Riemannian manifold and consider invariant, non-invariant and anti-invariant hypersurfaces of an almost r-paracontact Riemannian manifold with that connection.

Invariant Hypersurfaces in holomorphic Dynamics

We prove the following result, which is analogous to two theorems, one due to Kodaira and Krasnov and another one due to Jouanolou and Ghys. Let M be a compact complex manifold and f a dominant endomorphism of M. If there exist k totally invariant irreducible hypersurfacesWi M, with k > dim(M)+h1,1(M) then f preserves a nontrivial meromorphic fibration. We then study the case where f is a meromorphic map.

HYPERSURFACES OF ALMOST γ-PARACONTACT RIEMANNIAN MANIFOLD ENDOWED WITH SEMI-SYMMETRIC METRIC CONNECTION

We define a semi-symmetric metric connection in an almost <TEX>$\gamma$</TEX>-paracontact Riemannian manifold and we consider invariant, non-invariant and anti-invariant hypersurfaces of an almost <TEX>$\gamma$</TEX>-paracontact Riemannian manifold endowed with a semi-symmetric metric connection.

Hypersurfaces of an almost r-paracontact Riemannian Manifold Endowed with a Quarter Symmetric Non-metric Connection

We define a quarter symmetric non-metric connection in an almost r-paracontact Riemannian manifold and we consider invariant, non-invariant and anti-invariant hypersurfaces of an almost r-paracontact Riemannian manifold endowed with a quarter symmetric non-metric connection.

Hypersurfaces of an Almost r-Paracontact Riemannian Manifold with a Semi-Symmetric Non-Metric Connection

We define a semi-symmetric non-metric connection in an almost r-paracontact Riemannian manifold and consider invariant, non-invariant and anti-invariant hypersurfaces, respectively, of almost r-paracontact Riemannian manifold endowed with a semi-symmetric non-metric connection.

Darboux theory of integrability for polynomial vector fields in [formula omitted] taking into account the multiplicity at infinity

Darboux theory of integrability was established by Darboux in 1878, which provided a relation between the existence of first integrals and invariant algebraic hypersurfaces of vector fields in R n or C n with n ⩾ 2 . Jouanolou 1979 improved this theory to obtain rational first integrals via invariant algebraic surfaces using sophisticated tools of algebraic geometry. Recently in [J. Llibre, X. Zhang, Darboux theory of integrability in C n taking into account the multiplicity, J. Differential Equations, in press] this theory was improved taking into account not only the invariant algebraic hypersurfaces but also their multiplicity. In this paper we will show that if the hyperplane at infinity for a polynomial vector field in R n has multiplicity larger than 1, we can improve again the Darboux theory of integrability. We also show some difficulties for obtaining an extension of this result to polynomial vector fields in C n .

HYPERSURFACES OF ALMOST γ-PARACONTACT RIEMANNIAN MANIFOLD ENDOWED WITH A QUARTER SYMMETRIC METRIC CONNECTION

We define a quarter symmetric metric connection in an almost r-paracontact Riemannian manifold and we consider invariant, noninvariant and anti-invariant hypersurfaces of an almost r-paracontact Riemannian manifold endowed with a quarter symmetric metric connection.

Conservative stochastic differential equations: Mathematical and numerical analysis

We consider stochastic differential equations on the whole Euclidean space possessing a scalar invariant along their solutions. The stochastic dynamics therefore evolves on a hypersurface of the ambient space. Using orthogonal coordinate systems, we show the existence and uniqueness of smooth solutions of the Kolmogorov equation under some ellipticity conditions over the invariant hypersurfaces. If we assume, moreover, the existence of an invariant measure, we show the exponential convergence of the solution towards its average. In the second part, we consider numerical approximation of the stochastic differential equation, and show the convergence and numerical ergodicity of a class of projected schemes. In the context of molecular dynamics, this yields numerical schemes that are ergodic with respect to the microcanonical measure over isoenergy surfaces.

Darboux theory of integrability in [formula omitted] taking into account the multiplicity

Darboux in 1878 provided a theory on the existence of first integrals of polynomial systems based on the existence of sufficient invariant algebraic hypersurfaces, called now the Darboux theory of integrability. In 1979 Jouanolou successfully improved the Darboux theory of integrability characterizing the existence of rational first integrals, for this he used sophisticated tools of algebraic geometry. The aim of this paper is to improve the classical result of Darboux and the new one of Jouanolou taking into account the multiplicity of the invariant algebraic hypersurfaces. Additionally our proof of the improved result of Jouanolou is much simpler and elementary than the original one. Some examples show that the improved Darboux theory of integrability with multiplicity is much useful than the classical one.

On existence and uniqueness of the carrying simplex for competitive dynamical systems†

Certain dynamical models of competition are shown to have a unique invariant hypersurface Σ, having simple geometry and topology, such that every non-zero tractory is asymptotic to a trajectory in Σ.

Rational first integrals in the Darboux theory of integrability in [formula omitted

In 1979 Jouanolou showed that if the number of invariant algebraic hypersurfaces of a polynomial vector field in R n or C n of degree d is at least ( d + n − 1 n ) + n , then the vector field has a rational first integral. His proof used sophisticated tools of algebraic geometry. We provide an easy and elementary proof of Jouanolou's result using linear algebra.

Real hypersurfaces with η-parallel shape operator in complex two-plane Grassmannians

In this paper we give a characterisation of -invariant real hypersurfaces of type A; that is, a tube over a totally geodesic G2(ℂm+1) in complex two-plane Grassmannians G2(ℂm+2) or a ruled real hypersurface foliated by complex hypersurfaces which includes a maximal totally geodesic submanifold G2(ℂm+1) in G2(ℂm+2) in terms of η-parallel shape operator.

Invariant hypersurfaces for derivations in positive characteristic

Let A be an integral k -algebra of finite type over an algebraically closed field k of characteristic p > 0 . Given a collection D of k -derivations on A , that we interpret as algebraic vector fields on X = Spec ( A ) , we study the group spanned by the hypersurfaces V ( f ) of X invariant under D modulo the rational first integrals of D . We prove that this group is always a finite dimensional F p -vector space, and we give an estimate for its dimension. This is to be related to the results of Jouanolou and others on the number of hypersurfaces invariant under a foliation of codimension 1. As a application, given a k -algebra B between A p and A , we show that the kernel of the pull-back morphism Pic ( B ) → Pic ( A ) is a finite F p -vector space. In particular, if A is a UFD, then the Picard group of B is finite.

Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group ℍ n

In this article we study sets in the (2n + 1)-dimensional Heisenberg group ℍnwhich are critical points, under a volume constraint, of the sub-Riemannian perimeter associated to the distribution of horizontal vector fields in ℍn.We define a notion of mean curvature for hypersurfaces and we show that the boundary of a stationary set is a constant mean curvature (CMC) hypersurface. Our definition coincides with previous ones.

Complete and Stable O(p+1)×O(q+1)-Invariant Hypersurfaces with Zero Scalar Curvature in Euclidean Space ℝ p+q+2

We classify the zero scalar curvature O(p+1)×O(q+1)-invariant hypersurfaces in the euclidean space ℝp+q+2, p,q > 1, analyzing whether they are embedded and stable. The Morse index of the complete hypersurfaces show the existence of embedded, complete and globally stable zero scalar curvature O(p+1)×O(q+1)-invariant hypersurfaces in ℝp+q+2, p+q≥ 7, which are not homeomorphic to ℝp+q+1. Such stable examples provide counter-examples to a Bernstein-type conjecture in the stable class, for immersions with zero scalar curvature.

Pseudo-Einstein real hypersurfaces in complex two-plane Grassmannians

In this paper we give a complete classification of -invariant or Hopf pseudo-Einstein real hypersurfaces in complex two-plane Grassmannians G2(ℂm+2).

A floating body approach to Fefferman's hypersurface measure

The floating body approach to affine surface area is adapted to a holomorphic context providing an alternate approach to Fefferman's invariant hypersurface measure.

Regularity of area-minimizing surfaces in 3D polytopes and of invariant hypersurfaces inR n

In (the surface of) a convex polytope P 3 inR 4,an areaminimizing surface avoids the vertices of P and crosses the edges orthogonally. In a smooth Riemannian manifold M with a group of isometries G, an areaminimizing G-invariant oriented hypersurface is smooth (except for a very small singular set in high dimensions). Already in 3D, area-minimizing G-invariant unoriented surfaces can have certain singularities, such as three orthogonal sheets meeting at a point. We also treat other categories of surfaces such as rectifiable currents modulo v and soap films.

O(m) � O(n)-Invariant Minimal Hypersurfaces in $\mathbb{R}$ m+n

We classify the nonextendable immersed O(m) × O(n)-invariant minimal hypersurfaces in the Euclidean space \(\mathbb{R}\)m+n, m, n ≥ 3, analyzing also whether they are embedded or stable. We show also the existence of embedded, complete, stable minimal hypersurfaces in \(\mathbb{R}\)m+n, m + n ≥ 8, m, n ≥ 3 not homeomorphic to \(\mathbb{R}\)m+n−1 that are O(m) × O(n)-invariant.

Invariant analytic hypersurfaces in complex Lie groups

Suppose G is a complex Lie group and H is a closed complex subgroup of G. Let G′ denote the commutator subgroup of G. If there are no nonconstant holomorphic functions on G/H and H is not contained in any proper parabolic subgroup of G, then Akhiezer [2] asked whether every analytic hypersurface in G which is invariant under the right action of H is also invariant under the right action of G′. In this paper we answer a related question in two settings. Under the assumptions stated above we show that the orbits of the radical of G in G/H cannot be Cousin groups, provided G/H is Kähler. We also introduce an intermediate fibration of G/H induced by the holomorphic reduction of the radical orbits and resolve the related question in a situation arising from this fibration.