The sphere Sn+1 contains a simple family of constant mean curvature (CMC) hypersurfaces of the form Ct := Sp(cos t) × Sq(sin t) for p + q = n and t ∈ (0, π2 ) called the generalized Clifford hypersurfaces. This paper demonstrates that new, topologically non-trivial CMC hypersurfaces resembling a pair of neighbouring generalized Clifford tori connected to each other by small catenoidal bridges a...

We consider conformally flat hypersurfaces in four dimensional space forms with their associated Guichard nets and Lamé’s system of equations. We show that the symmetry group of the Lamé’s system, satisfying Guichard condition, is given by translations and dilations in the independent variables and dilations in the dependents variables. We obtain the solutions which are invariant under the acti...

Let D = A + λB be the para-Blaschke tensor of the immersion x, where λ is a constant, A and B are the Blaschke tensor and the Möbius second fundamental form of x. A hypersurface x : M 7→ S(1) in the unit sphere S(1) without umbilical points is called a para-Blaschke isoparametric hypersurface if the Möbius form Φ vanishes identically and all of its para-Blaschke eigenvalues are constants. In [1...

We asymptotically estimate from above the expected Betti numbers of random real hypersurfaces in smooth real projective manifolds. Our upper bounds grow as the square root of the degree of the hypersurfaces as the latter grows to infinity, with a coefficient involving the Kählerian volume of the real locus of the manifold as well as the expected determinant of random real symmetric matrices of ...

The theory of the isoptic curves is widely studied in the Euclidean plane E2 (see [1] and [13] and the references given there). The analogous question was investigated by the authors in the hyperbolic H2 and elliptic E2 planes (see [3], [4]), but in the higher dimensional spaces there is no result according to this topic. In this paper we give a natural extension of the notion of the isoptic cu...

This paper gives an asymptotic description of the Noether-Lefschetz locus for smooth projective hypersurfaces in P C of large degree. I prove that successive small codimensional components of this locus correspond to surfaces containing a small degree subvariety of dimension n. This result generalises the work of Green and Voisin for surfaces in PC containing a line and a conic. Résumé Cet arti...

In 2013, Abo and Wan studied Waring’s problem for systems of skew-symmetric forms and identified several defective systems. The cases of particular interest occur when a certain secant variety of a Segre-Grassmann variety is expected to fill the natural ambient space, but is actually a hypersurface. In these cases, one aims to obtain both a defining polynomial for these hypersurfaces along with...

Qualitative reasoning about physical processes is based on the notion of "quantity space" [Forbus, 1984a, 1984b]. The question is how to construct a quantity space for a particular physical process. One line of research is to establish a set of so called "landmark points" by selecting some values of the continuous physical parameters characterizing the physical process under consideration [Kuip...

In this paper, I give an explicit formula for the intersection indices of the Chern classes (defined earlier by the author) of an arbitrary reductive group with hypersurfaces. This formula has the following applications. First, it allows to compute explicitly the Euler characteristic of complete intersections in reductive groups thus extending the beautiful result by D. Bernstein and Khovanskii...

The techniques developed by Butscher (Gluing constructions amongst constant mean curvature hypersurfaces of Sn+1) for constructing constant mean curvature (CMC) hypersurfaces in Sn+1 by gluing together spherical building blocks are generalized to handle less symmetric initial configurations. The outcome is that the approximately CMC hypersurface obtained by gluing the initial configuration toge...