Abstract. Let (X m+1 , g) be a globally hyperbolic spacetime with Cauchy surface diffeomorphic to an open subset of R m . The Legendrian Low conjecture formulated by Natário and Tod says that two events x, y ∈ X are causally related if and only if the Legendrian link of spheres S x , S y whose points are light geodesics passing through x and y is nontrivial in the contact manifold of all light geodesics in X. The Low conjecture says that for m = 2 the events x, y are causally related if and only if S x , S y is non-trivial as a topological link. We prove the Low and the Legendrian Low conjectures. We also show that similar statements hold for any globally hyperbolic (X m+1 , g) such that a cover of its Cauchy surface is diffeomorphic to an open domain in R m .1. Introduction. The space N of non-parameterised future pointing null geodesics in a globally hyperbolic spacetime (X m+1 , g), m ≥ 2, has a natural structure of a contact (2m − 1)-manifold obtained by identifying N with the spherical cotangent bundle ST * M of a smooth spacelike Cauchy surface M m ⊂ X. Null geodesics passing through a point x ∈ X form a Legendrian (m − 1)-sphere S x ⊂ N called the sky of x. (Details and definitions may be found in § §3-4 below.)All skies in N are Legendrian isotopic. The situation is more interesting for links formed by pairs of disjoint skies. It was observed by Low [23] that the isotopy class of the link S x ⊔ S y may depend on whether the points x and y are causally related , that is, connected by a future pointing non-spacelike curve in X.It is not hard to show that all links S x ⊔ S y formed by skies of causally unrelated points belong to the same Legendrian isotopy class in N represented by a pair of fibres of ST * M. It is therefore natural to call S x and S y topologically unlinked (respectively, Legendrian unlinked ) if they are disjoint and the link S x ⊔ S y is smoothly (respectively, Legendrian) isotopic to a link in that 'trivial' isotopy class. It is also natural to ask whether the skies of causally related points are in some sense linked. This question was raised in different forms by Low [23]

It is shown that if the universal cover of a manifold M is an open manifold, then two different fibres of the spherical cotangent bundle S T M cannot be connected by a non-negative Legendrian isotopy. This result is applied to the study of causality in globally hyperbolic spacetimes. It is also used to strengthen a result of Eliashberg, Kim and Polterovich on the existence of a partial order on e Cont 0 .S T M /.

Introduction The semiconductor single-crystal CVD diamond (ob-tained from the gas phase during homoepitaxial deposi-tion) is a wide band gap semiconductor with a gap width of 5.5 eV. CVD diamond has unique characteristics-high mobility of charge carriers, high carrier saturation speed, high electric breakdown field, the greatest thermal conductivity, high radiation and chemical resistance. On a combination of properties the CVD diamond is superior to other wide band gap semiconductors and is considered a promising material for the creation of a new generation of high-power and high-frequency electronic devices. The main difficulty in realization of the potential of CVD diamond as an electronic material is the problem of creating charge carriers inside it. Compared with conventional semiconductors, dopants in diamond have deeper energy levels that significantly impede the activation of the do-pant (the degree of ionization of the dopant at room temperature is less than 1%). Thus, in order to create an acceptable level of conductivity, it is necessary to increase the level of doping, but in case of boron doping this leads to a decrease of carriers (holes) mobility in diamond. To solve the problem of boron doping of CVD diamond, an approach based on delta-doping technology is known. A thin layer of diamond heavily doped with boron (hav-ing a thickness of 1-2 nm and concentration of boron atoms higher than 5×10 20 cm-3) is formed inside an un-doped defect-free diamond of high quality. To achieve high electronic properties (obtaining high hole mobility and conductivity of the layer), it is necessary to realize sharp boundaries between the doped and undoped materials. Recently, this problem has been successfully solved [1, 2]. This report provides an overview of the results of studies on the growth of electronic-quality epitaxial layers of diamond, the production of heavily boron-doped layers and the study of their characteristics. Experiments The novel microwave plasma assisted CVD reactor for growth of nanometric boron delta-doped layers with ultra-sharp interfaces between doped/undoped materials was built in IAPRAS [1]. Fig. 1 shows a schematic of the reactor. The main features of the reactor are: 1) rapid gas switching; 2) laminar gas flow; 3) axial symmetric resonant mode-symmetric discharge; 4) slow growth of diamond 40-100 nm/h. We achieve rapid gas switching from one input gas to another by a home-made electronic switch. The residence time of the reactor is approximately 5 s. In developed reactor the diamond deposition regimes in which one obtains thin doped delta layers with thickness of 1-2 nm with concentrations of boron about 5·10 20 cm-3 were found. Typical parameters of the delta layer under these conditions are given in Fig. 2 for the SS6-1 sample, in which the boron concentration is 4.8·10 20 cm-3 and thickness is 1 nm. Measurement of the boron concentration in the grown samples was carried out by the secondary ion mass spectroscopy (SIMS) method using a time of flight SIMS setup (IONTOF TOF.SIMS-5). T...

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We report on building a novel chemical vapor deposition (CVD) reactor for diamond delta‐doping. The main features of our reactor are: a) the use of rapid gas switching system, (b) the reactor design providing the laminar gas flow. These features provide the creation of ultra‐sharp interfaces between doped and undoped material and minimize the prolonged ”tails” formation in the doping profile. It is proved by optical emission spectroscopy that gas switching time is not more than 10 seconds. Using the novel reactor we have grown the nanometer‐thin layers of boron doped diamond. The FWHM of boron concentration profile is about 2 nm which is proved by SIMS. It is shown that the both single delta‐layer and multiple delta‐layers could be grown using the novel CVD reactor. In principle, the reactor could be used for diamond delta doping with other dopants, like nitrogen, phosphorus etc. (© 2016 WILEY‐VCH Verlag GmbH &Co. KGaA, Weinheim)