Let (M, p) and (M̂, p̂) be the germs of real-analytic 1-infinite type hypersurfaces in C. We prove that any formal equivalence sending (M, p) into (M̂ , p̂) is formally parametrized (and hence uniquely determined by) its jet at p of a predetermined order depending only on (M, p). As an application, we use this to examine the local formal transformation groups of such hypersurfaces.

Let M be a compact Minimal hypersurface of the unit sphere S. In this paper we use a constant vector field on R to characterize the Clifford hypersurfaces S (√ l n ) × S mn ) , l + m = n, in S. We also study compact minimal Einstein hypersurfaces of dimension greater than two in the unit sphere and obtain a lower bound for first nonzero eigenvalue λ1 of its Laplacian operator.

The validity of global quadratic stability inequalities for uniquely regular area minimizing hypersurfaces is proved to be equivalent to the uniform positivity of the second variation of the area. Concerning singular area minimizing hypersurfaces, by a “quantitative calibration” argument we prove quadratic stability inequalities with explicit constants for all the Lawson’s cones, excluding six ...

(1.1) F̂ : x1 = 0, Ĝ : ξ2 = ξ 2 1 + x1 under a (C) smooth change of coordinates; Melrose’s argument also shows that all real analytic glancing hypersurfaces are equivalent to the above normal form by formal symplectic maps. It was proved by Oshima [6] for n ≥ 3 and by the second author [3] for n ≥ 2 that for some pairs of real analytic glancing hypersurfaces, the normal form cannot be achieved b...

A contact hypersurface in a Kähler manifold is a real hypersurface for which the induced almost contact metric structure determines a contact structure. We carry out a systematic study of contact hypersurfaces in Kähler manifolds. We then apply these general results to obtain classifications of contact hypersurfaces with constant mean curvature in the complex quadric Q = SOn+2/SOnSO2 and its no...

The gluing technique is used to construct hypersurfaces in Euclidean space having approximately constant prescribed mean curvature. These surfaces are perturbations of unions of finitely many spheres of the same radius assembled end-to-end along a line segment. The condition on the existence of these hypersurfaces is the vanishing of the sum of certain integral moments of the spheres with respe...

The potential energy surface (PES) of a molecule can be decomposed into equipotential hypersurfaces. We show in this article that the hypersurfaces are the wave fronts of a certain hyperbolic partial differential equation, a wave equation. It is connected with the gradient lines, or the steepest descent, or the steepest ascent lines of the PES. The energy seen as a reaction coordinate plays the...

Assuming minimal regularity assumptions on the data, we revisit the classical problem of finding isometric immersions into the Minkowski spacetime for hypersurfaces of a Lorentzian manifold. Our approach encompasses metrics having Sobolev regularity and Riemann curvature defined in the distributional sense, only. It applies to timelike, spacelike, or null hypersurfaces with arbitrary signature ...

In this note, we study properties of the gradient map of the isoparametric polynomial. For a given isoparametric hypersurface in sphere, we calculate explicitly the gradient map of its isoparametric polynomial which turns out many interesting phenomenons and applications. We find that it should map not only the focal submanifolds to focal submanifolds, isoparametric hypersurfaces to isoparametr...

We explicitly describe germs of strongly pseudoconvex non-spherical real-analytic hypersurfaces M at the origin in Cn+1 for which the group of local CR-automorphisms preserving the origin has dimension d0(M) equal to either n 2 − 2n + 1 with n ≥ 2, or n2 − 2n with n ≥ 3. The description is given in terms of equations defining hypersurfaces near the origin, written in the Chern-Moser normal form...