In this paper, we give a characterization of the finite multiplicity of a CR mapping between real analytic hypersurfaces. The finite multiplicity of a CR mapping was defined algebraically by Baouendi and Rothschild in [BR1] (see the definition below). We will prove that under certain conditions on hypersurfaces the finite multiplicity of a CR mapping is equivalent to that the preimage of the ma...

We give a proof of the monodromy conjecture relating poles motivic zeta functions with roots b-functions for isolated quasihomogeneous hypersurfaces, and more generally semi-quasihomogeneous hypersurfaces. also strange generalization allowing twist by certain differential forms.

In this paper we shall discuss hypersurfaces M of space forms of constant curvature; where curvature means one of the symmetric functions of curvature associated to the second fundamental form. The values of the constant will be chosen so that the linearized equation will be an elliptic equation onM . For example, for surfaces in 3 the two possible curvatures are the mean curvature H and the Ga...

We estimate from below the expected Betti numbers of real hypersurfaces taken at random in a smooth real projective n-dimensional manifold. These random hypersurfaces are chosen in the linear system of a large d-th power of a real ample line bundle equipped with a Hermitian metric of positive curvature. As for the upper bounds that we recently established, these lower bounds read as a product o...

The aim of this paper is to introduce a new family of monotone integral quantities associated with certain parabolic evolution equations for hypersurfaces, and to deduce from these some results about the limiting behaviour of the evolving hypersurfaces. A variety of parabolic equations for hypersurfaces have been considered. One of the earliest was the Gauss curvature flow, introduced in [Fi] a...

In Subsection 2.1 we have discussed uniqueness only in the special case in which f is discontinuous across a (d − 1)-dimensional hypersurface. The case when two or more such (d− 1)-dimensional hypersurfaces meet is significantly more involved and can lead to non-uniqueness of solutions for Filippov systems (see, e.g., [16]). In our setting, this situation arises at points in the set Eint define...

In this talk, first we introduce the classification of homogeneous hypersurfaces in some Hermitian symmetric spaces of rank 1 or rank 2. In particular, we give a full expression of the geometric structures for hypersurfaces in complex two-plane Grassmannians G2(C) or in complex hyperbolic twoplane Grassmannians G2(C). Next by using the isometric Reeb flow we give a complete classification for h...

We revisit Allendoerfer-Weil's formula for the Euler characteristic of embedded hypersurfaces in constant sectional curvature manifolds, first taking some time to re-prove it while demonstrating techniques [2] and then applying gain new understanding isoparametric hypersurfaces.

A Riemannian manifold satis es the axiom of 2-planes if at each point, there are su ciently many totally geodesic surfaces passing through that point. Real hypersurfaces in quaternionic space forms admit nice families of tangent planes, namely, totally real, half-quaternionic and quaternionic. Several de nitions of axiom of planes arise naturally when we consider such families of tangent planes...

V = {f | f is a homogeneous quintic polynomial of Z0, · · · , Z4}. We can verify that dimV = 126. Thus for any t ∈ P (V ) = CP , t is represented by a hypersurface. However, if two hypersurfaces differ by an element in Aut(CP ), then they are considered to be the same. Let D be the divisor in CP 125 characterizing the singular hypersurfaces in CP . Then the moduli space of X is M = CP \D/Aut(CP ).