Dimensional Hypersurface Research Articles

Petrov type I condition and dual fluid dynamics

Recently Lysov and Strominger [arXiv:1104.5502] showed that imposing Petrov type I condition on a $(p+1)$-dimensional timelike hypersurface embedded in a $(p+2)$-dimensional vacuum Einstein gravity reduces the degrees of freedom in the extrinsic curvature of the hypersurface to that of a fluid on the hypersurface, and that the leading-order Einstein constraint equations in terms of the mean curvature of the embedding give the incompressible Navier-Stokes equations of the dual fluid. In this paper we show that the non-relativistic fluid dual to vacuum Einstein gravity does not satisfy the Petrov type I condition at next order, unless additional constraint such as the irrotational condition is added. In addition, we show that this procedure can be inversed to derive the non-relativistic hydrodynamics with higher order corrections through imposing the Petrov type I condition, and that some second order transport coefficients can be extracted, but the dual "Petrov type I fluid" does not match the dual fluid constructed from the geometry of vacuum Einstein gravity in the non-relativistic limit. We discuss the procedure both on the finite cutoff surface via the non-relativistic hydrodynamic expansion and on the highly accelerated surface via the near horizon expansion.

Parabolic equations with dynamical boundary conditions and source terms on interfaces

We consider parabolic equations with mixed boundary conditions and domain inhomogeneities supported on a lower dimensional hypersurface, enforcing a jump in the conormal derivative. Only minimal regularity assumptions on the domain and the coefficients are imposed. It is shown that the corresponding linear operator enjoys maximal parabolic regularity in a suitable $$L^p$$ -setting. The linear results suffice to treat also the corresponding nondegenerate quasilinear problems.

On closed minimal hypersurfaces with constant scalar curvature in $$\mathbb{S}^7$$

We consider minimal closed hypersurfaces \({M \subset \mathbb{S}^7(1)}\) with constant scalar curvature. We prove that if M fulfills particular additional assumptions, then it is isoparametric. This gives a partial answer to the question made by S.-S. Chern about the pinching of the scalar curvature for closed minimal hypersurfaces in dimension 6.

Maximal ideal cycles over normal surface singularities of Brieskorn type

For normal two dimensional hypersurface singularities of Brieskorn type, concrete descriptions are given to both the fundamental cycle and the maximal ideal cycle on a star-shaped good resolution space. It is determined when these two cycles coincide.

Scalar curvature of minimal hypersurfaces in a sphere

We improve the well-known scalar curvature pinching theorem due to Peng–Terng for n (n⩽5)-dimensional minimal hypersurfaces to the case of arbitrary n. Precisely, if M is a closed and minimal hypersurface in a unit sphere Sn+1, then there exists a positive constant δ(n) depending only on n such that if n⩽S⩽n+δ(n), then S≡n, i.e., M is a Clifford torus Sk(kn)×Sn−k(n−kn), k=1,2,…,n−1.

DIRAC EQUATION ON A CURVED (2+1)-DIMENSIONAL HYPERSURFACE

Interest on (2+1)-dimensional electron systems has increased considerably after the realization of novel properties of graphene sheets, in which the behavior of electrons is effectively described by relativistic equations. Having this fact in mind, the following problem is studied in this work: when a spin-1/2 particle is constrained to move on a curved surface, is it possible to describe this particle without giving reference to the dimensions external to the surface? As a special case of this, a relativistic spin-1/2 particle which is constrained to move on a (2+1)-dimensional hypersurface of the (3+1)-dimensional Minkowskian spacetime is considered, and an effective Dirac equation for this particle is derived using the so-called thin layer method. Some of the results are compared with those obtained in a previous work by Burgess and Jensen.

Finite type hypersurfaces with divergent normal form

We consider the convergence problem for normal forms on finite type hypersurfaces in dimension two, constructed in Kolař (Math Res Lett 12:897–910, 2005). Two families of examples of hypersurfaces with divergent normal form are given. The results also show that the divergence phenomenon cannot be avoided by a modification of the normal form construction.

Regularity of weak solutions of $n$-dimensional H-systems

We use the method of Strzelecki [Calc. Var. 1 (2003)] to generalize the Bethuel theorem [C. R. Acad. Sci. Paris 314 (1992)] to $n$-dimensional H-systems. We prove that if $u$ is a parameterization of an $n$-dimensional hypersurface in $\mathbb R^{n+1}$, weakly satisfies the system $\Delta_n u = H(u) u_{x_1} \times \cdots \times u_{x_n}$ and additionally has $n-1$ weak derivatives in $L^{n/(n-1)}$, then $u$ is Hölder continuous. Furthermore it is continuous up to the boundary, whenever it has continuous trace. We also give an example showing that the structure of the H-system is relevant and that the assumption that $u$ has $n-1$ weak derivatives does not trivialize the problem.

On Three Dimensional Real Hypersurfaces in Complex Space Forms

On Three Dimensional Real Hypersurfaces in Complex Space Forms

ABEL–JACOBI MAPS FOR HYPERSURFACES AND NONCOMMUTATIVE CALABI–YAU'S

It is well known that the Fano scheme of lines on a cubic 4-fold is a symplectic variety. We generalize this fact by constructing a closed (2n - 4)-form on the Fano scheme of lines on a (2n - 2)-dimensional hypersurface Yn of degree n. We provide several definitions of this form — via the Abel–Jacobi map, via Hochschild homology, and via the linkage class — and compute it explicitly for n = 4. In the special case of a Pfaffian hypersurface Yn we show that the Fano scheme is birational to a certain moduli space of sheaves of a (2n - 4)-dimensional Calabi–Yau variety X arising naturally in the context of homological projective duality, and that the constructed form is induced by the holomorphic volume form on X. This remains true for a general non-Pfaffian hypersurface but the dual Calabi–Yau becomes noncommutative.

THE HIERARCHY PROBLEM, RADION MASS, LOCALIZATION OF GRAVITY AND 4D EFFECTIVE NEWTONIAN POTENTIAL IN STRING THEORY ON S1/Z2

In this paper, we present a systematical study of braneworlds of string theory on S1/Z2. In particular, starting with the toroidal compactification of the Neveu–Schwarz/Neveu–Schwarz sector in D + d dimensions, we first obtain an effective D-dimensional action, and then compactify one of the D - 1 spatial dimensions by introducing two orbifold branes as its boundaries. We divide the whole set of the gravitational and matter field equations into two groups, one holds outside the two branes, and the other holds on them. By combining the Gauss–Codacci and Lanczos equations, we write down explicitly the general gravitational field equations on each of the two branes, while using distribution theory we express the matter field equations on the branes in terms of the discontinuities of the first derivatives of the matter fields. Afterwards, we address three important issues: (i) the hierarchy problem; (ii) the radion mass; and (iii) the localization of gravity, the four-dimensional Newtonian effective potential and the Yukawa corrections due to the gravitational high-order Kaluza–Klein (KK) modes. The mechanism of solving the hierarchy problem is essentially the combination of the large extra dimension and warped factor mechanisms together with the tension coupling scenario. With very conservative arguments, we find that the radion mass is of the order of 10-2 GeV. The gravity is localized on the visible brane, and the spectrum of the gravitational KK modes is discrete and can be of the order of TeV. The corrections to the four-dimensional Newtonian potential from the higher order of gravitational KK modes are exponentially suppressed and can be safely neglected in current experiments. In an appendix, we also present a systematical and pedagogical study of the Gauss–Codacci equations and Israel's junction conditions across a (D - 1)-dimensional hypersurface, which can be either spacelike or timelike.

Brane cosmology and motion of test particles in five-dimensional warped product spacetimes

In the braneworld scenario ordinary standard model matter and on-gravitational fields are confined by some trapping mechanism to the 4-dimensional universe constituting the D3-branes which are embedded in a (4 + n)-dimensional manifold referred to as the bulk (n being the number of extra dimensions). The notion of particle confinement is necessary for theories with non-compact extra dimensions, otherwise, the particles would escape from our 4-dimensional world along unseen directions. In this paper, we have considered a five-dimensional warped product space-time having an exponential warping function which depends both on time as well as on the extra coordinates and a non-compact fifth dimension. Further, we have assumed that the lapse function may either be a constant or a function of both time and of the extra coordinates, and have studied the nature of the geodesics of test particles and analyzed the conditions of stability in this geometrical framework. We have also discussed the possible cosmology of the corresponding (3 + 1)-dimensional hypersurfaces.

Topology of real cubic fourfolds

A solution of the problem of topological classification of real cubic fourfolds is given. It is proven that the real locus of a real non-singular cubic fourfold is diffeomorphic either to a connected sum ℝP4 #i(S2 × S2)#j(S1 × S3) or to a disjoint union ℝP4 ⊔ S4. La voie la plus courte et la meilleure entre deux vérités de domaine réel passe souvent par le domaine imaginaire.J. S. Hadamard, Essai sur la psychologie de l'invention dans le domainemathématique, Edition Gaithier-Villars, 1975, p. 114.

Rational Curves on Smooth Cubic Hypersurfaces

We prove that the space of rational curves of a fixed degree on any smooth cubic hypersurface of dimension at least four is irreducible and of the expected dimension. Our methods also show that the space of rational curves of a fixed degree on a general hypersurface in of degree 2d ≤ min (n + 4, 2n − 2) and dimension at least three is irreducible and of the expected dimension.

On Semiparallel and Weyl-semiparallel Hypersurfaces of Kaehler Manifolds

We study on semiparallel and Weyl semiparallel Sasakian hypersurfaces of Kaehler manifolds. We prove that a (2n + 1)-dimensional Sasakian hypersurface M of a (2n+2)-dimensional Kaehler manifold is semiparallel if and only if it is totally umbilical with unit mean curvature, if dimM = 3 and is a Calabi-Yau manifold, then is flat at each point of M. We also prove that such a hypersurface M is Weyl-semiparallel if and only if it is either an -Einstein manifold or semiparallel. We also investigate the extended classes of semiparallel and Weyl semiparallel Sasakian hypersurfaces of Kaehler manifolds.

Low-type submanifolds of real space forms via the immersions by projectors

We undertake a comprehensive study of submanifolds of low Chen-type (1, 2, or 3) in non-flat real space forms, immersed into a suitable (pseudo) Euclidean space of symmetric matrices by projection operators. Some previous results for submanifolds of the unit sphere (obtained in [A. Ros, Eigenvalue inequalities for minimal submanifolds and P-manifolds, Math. Z. 187 (1984) 393–404; M. Barros, B.Y. Chen, Spherical submanifolds which are of 2-type via the second standard immersion of the sphere, Nagoya Math. J. 108 (1987) 77–91; I. Dimitrić, Spherical hypersurfaces with low type quadric representation, Tokyo J. Math. 13 (1990) 469–492; J.T. Lu, Hypersurfaces of a sphere with 3-type quadric representation, Kodai Math. J. 17 (1994) 290–298]) are generalized and extended to real projective and hyperbolic spaces as well as to the sphere. In particular, we give a characterization of 2-type submanifolds of these space forms with parallel mean curvature vector. We classify 2-type hypersurfaces in these spaces and give two sets of necessary conditions for a minimal hypersurface to be of 3-type and for a hypersurface with constant mean curvature to be mass-symmetric and of 3-type. These conditions are then used to classify such hypersurfaces of dimension n ⩽ 5 . For example, the complete minimal hypersurfaces of the unit sphere S n + 1 which are of 3-type via the immersion by projectors are exactly the 3-dimensional Cartan minimal hypersurface SO ( 3 ) / ( Z 2 × Z 2 ) and the Clifford minimal hypersurfaces M k , n − k for n ≠ 2 k . An interesting characterization of horospheres in R H n + 1 is also obtained.

Black Holes in the Dilatonic Einstein-Gauss-Bonnet Theory in Various Dimensions. II: Asymptotically AdS Topological Black Holes

We study asymptotically AdS topological black hole solutions with k =0 (plane symmetric) in the Einstein gravity with Gauss-Bonnet term, the dilaton and a “cosmological constant” in various dimensions. We derive the field equations for suitable ansatz for general D dimensions. We determine the parameter regions including dilaton couplings where such solutions exist and construct black hole solutions of various masses numerically in D =4 , 5, 6 and 10 dimensional spacetime with (D − 2)-dimensional hypersurface of zero curvature. Subject Index: 121, 122, 454

Higher order Willmore hypersurfaces in Euclidean space

Let x: M n → R n+1 be an n(≥ 2)-dimensional hypersurface immersed in Euclidean space R n+1. Let σ i (0 ≤ i ≤ n) be the ith mean curvature and Q n = Σ i=0 n (−1)i+1( i n )σ 1 n−i σ i . Recently, the author showed that W n (x) = ∫ M Q n dM is a conformal invariant under conformal group of R n+1 and called it the nth Willmore functional of x. An extremal hypersurface of conformal invariant functional W n is called an nth order Willmore hypersurface. The purpose of this paper is to construct concrete examples of the 3rd order Willmore hypersurfaces in R 4 which have good geometric behaviors. The ordinary differential equation characterizing the revolutionary 3rd Willmore hypersurfaces is established and some interesting explicit examples are found in this paper.

RIGIDITY OF MINIMAL SUBMANIFOLDS WITH FLAT NORMAL BUNDLE

Abstract. Let M n be a complete immersed super stable minimal sub-manifold in R n + p with °at normal bundle. We prove that if M has flnitetotal L 2 norm of its second fundamental form, then M is an a–ne n -plane. We also prove that any complete immersed super stable minimalsubmanifold with °at normal bundle has only one end. 1. IntroductionLet M be an n -dimensional complete minimal submanifold in R n + p . When n = 2 and p = 1, do Carmo and Peng [3], Fischer-Colbrie and Schoen [5]independently showed that the only complete stable minimal surface is a plane.Recall that a minimal submanifold is stable if the second variation of its volumeis always nonnegative for any normal variation with compact support. Forhigher dimensional minimal hypersurfaces, do Carmo and Peng [4] generalizedthe result mentioned as above. We will denote by A the second fundamentalform of M .Theorem ([4]). Let M n be a complete stable minimal hypersurface in R n +1 satisfying R M jAj 2 dv < 1. Then M must be a hyperplane.

Black Holes in the Dilatonic Einstein-Gauss-Bonnet Theory in Various Dimensions. I: -- Asymptotically Flat Black Holes --

We study spherically symmetric, asymptotically flat black hole solutions in the low-energy effective heterotic string theory, which is the Einstein gravity with Gauss-Bonnet term and the dilaton, in various dimensions. We derive the field equations for suitable ansatz for general D dimensions and construct black hole solutions of various masses numerically in D=4,5,6 and 10 dimensional spacetime with (D-2)-dimensional hypersurface with positive constant curvature. A detailed comparison with the non-dilatonic solutions is made. We also examine the thermodynamic properties of the solutions. It is found that the dilaton has significant effects on the black hole solutions, and we discuss physical consequences.