In this paper, we consider minimal hypersurfaces in the product space Hn×R. We study the relation between the notions of finite total curvature and index of the stability operator. We study examples of rotation hypersurfaces and hypersurfaces invariant under hyperbolic translations; they serve as counterexamples and are useful barriers for many geometric problems.1

The first part of this paper considers higher order CR invariants of three dimensional hypersurfaces of finite type. Using a full normal form we give a complete characterization of hypersurfaces with trivial local automorphism group, and analogous results for finite groups. The second part considers hypersurfaces of finite Catlin multitype, and the Kohn-Nirenberg phenomenon in higher dimensions...

This paper studies the Kohn-Nirenberg phenomenon existence of weakly pseudoconvex but locally nonconvexifiable hypersurfaces. We give a characterization of such hypersurfaces in terms of a generalized model, which captures behaviour of the hypersurface also in the complex nontangential direction. As an application we obtain a new class of nonconvexifiable pseudoconvex hypersurfaces with convex ...

While studying the existence of closed geodesics and minimal hypersurfaces in compact manifolds, the concept of width was introduced in different contexts. Generally, the width is realized by the energy of the closed geodesics or the volume of minimal hypersurfaces, which are found by the Minimax argument. Recently, Marques and Neves used the p-width to prove the existence of infinite many mini...

We introduce in this paper the concept of tropical mirror hypersurfaces and we prove a complex tropical localization Theorem which is a version of Kapranov's Theorem [K-00] in tropical geometry. We give a geometric and a topological equivalence between coamoebas of complex algebraic hypersurfaces de ned by a maximally sparse polynomial and coamoebas of maximally sparse complex tropical hypersur...

We determine all stable constant mean curvature hypersurfaces in a wide class of complete Riemannian manifolds having a foliation whose leaves are umbilical hypersurfaces. Among the consequences of this analysis we obtain all the stable constant mean curvature hypersurfaces in many nonsimply connected hyperbolic space forms. Mathematics Subject Classification (1991). Primary 53A10; Secondary 49...

Abstract. In 1968, Simons [9] introduced the concept of index for hypersurfaces immersed into the Euclidean sphere S. Intuitively, the index measures the number of independent directions in which a given hypersurface fails to minimize area. The earliest results regarding the index focused on the case of minimal hypersurfaces. Many such results established lower bounds for the index. More recent...

We observe that nonzero Gromov-Witten invariants with marked point constraints in a closed symplectic manifold imply restrictions on the homology classes that can be represented by contact hypersurfaces. As a special case, contact hypersurfaces must always separate if the symplectic manifold is uniruled. This removes a superfluous assumption in a result of G. Lu [Lu00], thus implying that all c...

É. Cartan proved that conformally flat hypersurfaces in Sn+1 for n > 3 have at most two distinct principal curvatures and locally envelop a one-parameter family of (n − 1)-spheres. We prove that the Gauss-Codazzi equation for conformally flat hypersurfaces in S4 is a soliton equation, and use a dressing action from soliton theory to construct geometric Ribaucour transforms of these hypersurface...

It is our purpose to study curvature structures of compact hypersurfaces in the unit sphere S(1). We proved that the Riemannian product S( √ 1 − c2) ×Sn−1(c) is the only compact hypersurfaces in S(1) with infinite fundamental group, which satisfy r ≥ n−2 n−1 and S ≤ (n − 1)n(r−1)+2 n−2 + n−2 n(r−1)+2 , where n(n − 1)r is the scalar curvature of hypersurfaces and c = n−2 nr . In particular, we o...