We introduce new biholomorphic invariants for real-analytic hypersurfaces in C and show how they can be used to show that a hypersurface possesses few automorphisms. We give conditions, in terms of the new invariants, guaranteeing that the stability group is finite, and give (sharp) bounds on the cardinality of the stability group in this case. We also give a sufficient condition for the stabil...

In this paper, we obtain a criterion of boundedness maximal operators associated with smooth hypersurfaces. Also, compute the exact value index such arbitrary convex analytic hypersurfaces in case where Varchenko height hypersurface is greater than two. We for degenerate hypersurfaces, i.e., satisfying assumptions classical Hartman–Nirenberg theorem. The obtained results justify Stein–Iosevich–...

In this paper, we study the close spacelike hypersurfaces in de Sitter space. Using Bonnet-Myer’s theorem, we prove a rigidity theorem for spacelike hypersurfaces without the constancy condition on the mean curvature or the scalar curvature. M.S.C. 2010: 53C40, 53B30.

We study the birational properties of hypersurfaces in products of projective spaces. In the case of hypersurfaces in P × P, we describe their nef, movable and effective cones and determine when they are Mori dream spaces. Using this, we give new simple examples of non-Mori dream spaces and analogues of Mumford’s example of a strictly nef line bundle which is not ample.

We derive a formula for the first variation of horizontal perimeter measure for C2 hypersurfaces of completely general sub-Riemannian manifolds, allowing for the existence of characteristic points. When the manifold admits dilations, we establish a sub-Riemannian Minkowski formula. For C2 hypersurfaces in vertically rigid sub-Riemannian manifolds we also produce a second variation formula for v...

In this paper we find new examples of Riemannian manifolds with nonspherical apparent horizon, in dimensions four and above. More precisely, for any n, m ≥ 1, we construct asymptotically flat, scalar flat Riemannian manifolds containing smooth outermost minimal hypersurfaces with topology Sn × S. In the context of general relativity these hypersurfaces correspond to outermost apparent horizons ...

We use slicing by nongeneric pencils of hypersurfaces and prove a new theorem of Lefschetz type for singular non compact spaces, at the homotopy level. As applications, we derive results on the topology of the fibres of polynomial functions or the topology of complements of hypersurfaces in Cn.

Almost-toric projective hypersurfaces could be parameterized by n+2 monomials in n variables multiplying univariate polynomials in an extra variable. In this paper we give a combinatorial description of the Newton polytope (actually it’s a polygon) of the defining polynomial of almosttoric hypersurfaces.

This paper studies local geometry of hypersurfaces of finite multitype. Catlin’s definition of multitype is applied to a general smooth hypersurface in C. We prove biholomorphic equivalence of models in dimension three and describe all biholomorphisms between such models. A finite constructive algorithm for computing multitype is described. Analogous results for decoupled hypersurfaces are given.

We study rotational hypersurfaces with constant 2-mean curvature in Rn+1. derive the ODE for generating curves of such hypersurfaces, and find an integral expression inverse function solution.