Flat Hypersurfaces Research Articles

Local equivalence problem for Levi flat hypersurfaces

In this paper we consider germs of smooth Levi flat hypersurfaces, under the following notion of local equivalence: S 1 ∼ S 2 if their one-sided neighborhoods admit a biholomorphism smooth up to the boundary. We introduce a simple invariant for this relation, which allows us to prove some characterizations of triviality (i.e. equivalence to the hyperplane). Then, we employ the same invariant to construct infinitely many non-trivial classes, including an infinite family of non-equivalent hypersurfaces which are almost everywhere analytic.

Foliation of the Kottler–Schwarzschild–de Sitter spacetime by flat spacelike hypersurfaces

There exist Kruskal like coordinates for the Reissner–Nordstrom (RN) black hole spacetime which are regular at coordinate singularities. Non-existence of such coordinates for the extreme RN black hole spacetime has already been shown. Also the Carter coordinates available for the extreme case are not manifestly regular at the coordinate singularity, therefore, a numerical procedure was developed to obtain free fall geodesics and flat foliation for the extreme RN black hole spacetime. The Kottler–Schwarzschild–de Sitter (KSSdS) spacetime geometry is similar to the RN geometry in the sense that, like the RN case, there exist non-singular coordinates when there are two distinct coordinate singularities. There are no manifestly regular coordinates for the extreme KSSdS case. In this paper foliation of all the cases of the KSSdS spacetime by flat spacelike hypersurfaces is obtained by introducing a non-singular time coordinate.

On the characteristic direction of real hypersurfaces in and a symmetry result

Abstract In this paper we show the following property of a non Levi flat real hypersurface in : if the unit characteristic direction T is a geodesic, then it is an eigenvector of the second fundamental form and the relative eigenvalue is constant. As an application we prove a symmetry result of Alexandrov type for compact hypersurfaces in with positive constant Levi mean curvature.

Generalized Voronoi Tessellation as a Model of Two-dimensional Cell Tissue Dynamics

Voronoi tessellations have been used to model the geometric arrangement of cells in morphogenetic or cancerous tissues, however, so far only with flat hyper-surfaces as cell-cell contact borders. In order to reproduce the experimentally observed piecewise spherical boundary shapes, we develop a consistent theoretical framework of multiplicatively weighted distance functions, defining generalized finite Voronoi neighborhoods around cell bodies of varying radius, which serve as heterogeneous generators of the resulting model tissue. The interactions between cells are represented by adhesive and repelling force densities on the cell contact borders. In addition, protrusive locomotion forces are implemented along the cell boundaries at the tissue margin, and stochastic perturbations allow for non-deterministic motility effects. Simulations of the emerging system of stochastic differential equations for position and velocity of cell centers show the feasibility of this Voronoi method generating realistic cell shapes. In the limiting case of a single cell pair in brief contact, the dynamical nonlinear Ornstein-Uhlenbeck process is analytically investigated. In general, topologically distinct tissue conformations are observed, exhibiting stability on different time scales, and tissue coherence is quantified by suitable characteristics. Finally, an argument is derived pointing to a tradeoff in natural tissues between cell size heterogeneity and the extension of cellular lamellae.

Nonconformally flat initial data for binary compact objects

A new method is described for constructing initial data for a binary neutron-star (BNS) system in quasi-equilibrium circular orbit. Two formulations for non-conformally flat data, waveless (WL) and near-zone helically symmetric (NHS), are introduced; in each formulation, the Einstein-Euler system, written in 3+1 form on an asymptotically flat spacelike hypersurface, is exactly solved for all metric components, including the spatially non-conformally flat potentials, and for irrotational flow. A numerical method applicable to both formulations is explained with an emphasis on the imposition of a spatial gauge condition. Results are shown for solution sequences of irrotational BNS with matter approximated by parametrized equations of state that use a few segments of polytropic equations of state. The binding energy and total angular momentum of solution sequences computed within the conformally flat -- Isenberg-Wilson-Mathews (IWM) -- formulation are closer to those of the third post-Newtonian (3PN) two point particles up to the closest orbits, for the more compact stars, whereas sequences resulting from the WL/NHS formulations deviate from the 3PN curve even more for the sequences with larger compactness. We think it likely that this correction reflects an overestimation in the IWM formulation as well as in the 3PN formula, by $\sim 1$ cycle in the gravitational wave phase during the last several orbits. The work suggests that imposing spatial conformal flatness results in an underestimate of the quadrupole deformation of the components of binary neutron-star systems in the last few orbits prior to merger.

A Jang equation approach to the Penrose inequality

We introduce a generalized version of the Jang equation, designedfor the general case of the Penrose Inequality in the setting ofan asymptotically flat space-like hypersurface of a spacetimesatisfying the dominant energy condition. The appropriateexistence and regularity results are established in the specialcase of spherically symmetric Cauchy data, and are applied to givea new proof of the general Penrose Inequality for these data sets.When appropriately coupled with an inverse mean curvature flow,analogous existence and regularity results for the associatedsystem of equations in the nonspherical setting would yield aproof of the full Penrose Conjecture. Thus it remains as animportant and challenging open problem to determine whether thissystem does indeed admit the desired solutions.

Flat lightlike hypersurfaces in Lorentz–Minkowski 4-space

The lightlike hypersurfaces in Lorentz–Minkowski space are of special interest in Relativity Theory. In particular, the singularities of these hypersurfaces provide good models for the study of different horizon types. We introduce the notion of flatness for these hypersurfaces and study their singularities. The classification result asserts that a generic classification of flat lightlike hypersurfaces is quite different from that of generic lightlike hypersurfaces.

Classification results for biharmonic submanifolds in spheres

We study biharmonic submanifolds of the Euclidean sphere that satisfy certain geometric properties. We classify: (i) the biharmonic hypersurfaces with at most two distinct principal curvatures; (ii) the conformally flat biharmonic hypersurfaces. We obtain some rigidity results for pseudoumbilical biharmonic submanifolds of codimension 2 and for biharmonic surfaces with parallel mean curvature vector field. We also study the type, in the sense of B-Y. Chen, of compact proper biharmonic submanifolds with constant mean curvature in spheres.

A note on the restriction theorem and geometry of hypersurfaces

A necessary condition is established for the optimal $(L^p,L^2)$ restriction theorem to hold on a hypersurface $S$, in terms of its Gaussian curvature. For some classes of flat hypersurface we give sharp thresholds for the range of admissible exponents $p$, depending on the specific geometry.

Complete hypersurfaces in a Euclidean space [formula omitted] with constant mth mean curvature

In this paper, we consider n-dimensional oriented complete hypersurfaces with constant mth mean curvature of a Euclidean space R n + 1 . We characterize the hypersurface S k ( c ) × R n − k in a Euclidean space R n + 1 and show that generalized Yau conjecture is true for the class of oriented compact locally conformally flat hypersurfaces with positive constant mth mean curvature of a Euclidean space R n + 1 . When m = 2 , our results reduce to the results of Q.-M. Cheng [Q.M. Cheng, Complete hypersurfaces in a Euclidean space R n + 1 with constant scalar curvature, Indiana Univ. Math. J. 51 (2002) 53–68].

Uniqueness of flat spherically symmetric spacelike hypersurfaces admitted by spherically symmetric static spacetimes

It is known that spherically symmetric static spacetimes admit a foliation by flat hypersurfaces. Such foliations have explicitly been constructed for some spacetimes, using different approaches, but none of them have proved or even discussed the uniqueness of these foliations. The issue of uniqueness becomes more important due to suitability of flat foliations for studying black hole physics. Here, flat spherically symmetric spacelike hypersurfaces are obtained by a direct method. It is found that spherically symmetric static spacetimes admit flat spherically symmetric hypersurfaces, and that these hypersurfaces are unique up to translation under the timelike Killing vector. This result guarantees the uniqueness of flat spherically symmetric foliations for such spacetimes.

Conformally flat Lorentzian hypersurfaces in the conformal compactification of Lorentz space

We study conformally flat Lorentzian hypersurfaces in the conformal compactification of Lorentz space R 1 n + 1 , which is the projectivized light cone R ̂ 1 n + 1 ⊂ R P n + 2 induced from R 2 n + 3 . We establish a Lorentzian version of the local classification theorem of Cartan, in terms of branched channel hypersurfaces for n ≥ 4 , and for n = 3 , in terms of the conformal fundamental forms. For hypersurfaces whose shape operator has complex eigenvalues, we give a necessary condition for being conformally flat in terms of local integrability of distributions.

Quasi Conformally at Sasakian Hypersurface of the Generalized Recurrent Kahlerian Manifold

In the present paper we consider a Quasi-Conformally flat Sasakian hypersurface of the generalized recurrent K¨ahlerian manifold and study the conditions under which it is η-Einstein. We also study the conditions for the scalar curvature to be constant, when its ricci tensor is cyclic parallel and η-parallel.

Komplexe Analysis

The workshop Komplexe Analysis , organised by Jean-Pierre Demailly (Grenoble), Klaus Hulek (Hannover) and Thomas Peternell (Bayreuth) was held August 27th–September 2nd, 2006. This meeting was well attended with over 50 participants from Europe, US, Israel and Far East. The participants included several leaders of the field as well as many young (non-tenured) researchers, including four doctoral students. The aim of the meeting was to present recent important results in several complex variables and complex geometry with a particular emphasis on topics linking different areas of the field, as well as to discuss new directions and open problems. Altogether there were eighteen talks of 60 minutes each, a programme which left sufficient time for informal discussions and joint work on research projects. One of the topics under discussion was the classification theory of higher dimensional varieties. The key note lecture was given by J. McKernan who announced a proof of the main conjecture in Mori theory, which says that all smooth projective varieties of general type have a minimal and a canonical model. This is equivalent to finite generation of the canonical ring. Outside the official programme McKernan also explained some of the technical details of the proof. In his talk S. Kovács gave a proof of a conjecture of Viehweg on families of canonically polarized varieties over a surface. On the other side of the classification problem, the geometry of Fano manifolds and their classification also played an important role. P. Jahnke lectured on recent progress in the classification of Fano manifolds and A. V. Pukhlikov spoke on the birational geometry of these varieties. Finally, J.-M. Hwang discussed the problem which Fano manifolds can be rigid targets of holomorphic maps and M. Brion presented his work on the structure of log homogeneous pairs. Mirror symmetry and related topics remain an important aspect of the field. This was addressed in three talks. B. Siebert spoke on tropical games and mirror symmetry. Tropical geometry was also the topic of A. Gathmann's talk who discussed recent progress in enumerative geometry in connection with tropical methods. On the other end of the temperature scale D. van Straten explained arctic monodromy calculations. Hyperbolicity was the central theme of the talks of E. Rousseau, who proved hyperbolicity type results for generic hypersurfaces of sufficiently high degree in 4 -dimensional projective space, and in the programmatic lecture by M. McQuillan. The talks by Ph. Eyssidieux on singular Kähler–Einstein metrics and by T. Ohsawa on Levi flat hypersurfaces in complex manifolds were predominantly analytic in nature. More classically geometric topics were discussed by F. Catanese (slope of Kodaira fibrations), F. Bogomolov (symmetric tensors and the geometry of secant varieties) and P. M. H. Wilson (geometry of Kähler moduli). Finally, A. Sommese discussed techniques on how to decompose certain varieties into irreducible components. These questions were motivated by engineering problems and their solution has many practical applications.

On the Displacement Rigidity of Levi Flat Hypersurfaces—The Case of Boundaries of Disc Bundles Over Compact Riemann Surfaces

Non-existence theorems for Levi flat hypersurfaces have found great interest in the literature. The question next to this that has to be asked is, when existing Levi flat hypersurfaces are at least rigid under deformations. Here, the case of boundaries of disc bundles over certain compact Riemann surfaces is considered.

CONFORMALLY FLAT HYPERSURFACES WITH CYCLIC GUICHARD NET

We classify the 3-dimensional conformally flat hypersurfaces with cyclic principal Guichard net. The orthogonal surfaces of the cyclic system turn out to be linear Weingarten surfaces in suitable ambient space forms and we provide explicit parametrizations for the conformally flat hypersurfaces.

The Classification of Static Electro–Vacuum Space–Times Containing an Asymptotically Flat Spacelike Hypersurface with Compact Interior

We show that static electro–vacuum black hole space–times containing an asymptotically flat spacelike hypersurface with compact interior and with both degenerate and non–degenerate components of the event horizon do not exist. This is done by a careful study of the near-horizon geometry of degenerate horizons, which allows us to eliminate the last restriction of the static electro-vacuum no-hair theory.

Conformally flat Lorentzian hypersurfaces and curved flats

Conformally flat Lorentzian hypersurfaces and curved flats

On the Complement of Levi-Flats in Kähler Manifolds of Dimension ≥ 3

AbstractApplying the L2 method of solving the -equation, it is shown that compact Kähler manifolds of dimension ≥ 3 admit no Levi flat real analytic hypersurfaces whose complements are Stein.

Conformally flat Lorentzian hypersurfaces and curved flats

Conformally flat Lorentzian hypersurfaces and curved flats