Hypersurfaces In Rn Research Articles

Nonlinear partial differential systems on Riemannian manifolds with their geometric applications

We make the first study of how the existence of (essential) positive supersolutions of nonlinear degenerate partial differential equations on a manifold affects the topology, geometry, and analysis of the manifold. For example, for surfaces in R3 we prove a Bernstein-type theorem that generalizes and unifies three distinct theorems. In higher dimensions, we provide topological obstructions for a minimal hypersurface in Rn+1 to admit an essential positive supersolution. This immediately yields information about the Gauss map of complete minimal hypersurfaces in Rn+1. By coping with a wider class of nonlinear partial differential equations that are involved with (p)-harmonic maps and (p)-superstrongly unstable manifolds, we derive information on the regularity of minimizers, homotopy groups, and solutions to Dirichlet problems, from the existence of essential positive supersolutions.

Hypersurfaces in ℝn and critical points in their external region

In this paper we study the hypersurfaces \(M^n \) given as connected compact regular fibers of a differentiable map \(f:\mathbb{R}^{n + 1} \to \mathbb{R}\), in the cases in which \(f\) has finitely many nondegenerate critical points in the unbounded component of \(\mathbb{R}^{n + 1} - M^n \).

Erratum to "Bernstein-type theorems in hypersurfaces with constant mean curvature" [An Acad Bras Cienc 72(2000): 301-310

2. A RESULT ON NODAL DOMAINS In this section we prove a result on the nodal domains of |φ| which will be needed in our proof of main theorems. We first need to recall the definition of nodal domains. Definition. An open domain D is called the nodal domain of a function f if f (x) = 0 for x ∈ intD and vanishes on the boundary of ∂D. We denote by N(f ) the number of disjoint bounded nodal domains of f . Now we have the following lemma which follows directly from Proposition 2.2 below. We want to thank the referee who provided the clearer proof of Proposition 2.2. Lemma 2.1. Let M be a hypersurface in Rn+1 with constant mean curvature H . Then ind(M) ≥ N(|φ|). (2.1)

Maximal Functions over Hypersurfaces with Flat Points

For n≥4, let S be a flat hypersurface in Rn, and let dμ=ψdσ, where ψ∈C∞0(Rn) and σ is the surface area measure on S. Then the maximal functions Mf associated to S and μ by Mf(x)=supt>0|∫Sf(x−tξ)dμ(ξ)|, f∈S(Rn), are bounded on certain Orlicz spaces LΦ(Rn).

On hypersurfaces inR n+1 with integral bounds on curvature

We show that the Lpnorm of the second fundamental form of hypersurfaces inRn+1is very coercive in the GMT setting of Gauss graphs currents, when p exceeds the dimension n. A compactness result for immersed hypersurfaces and its application to a variational problem are provided.

Recognizing unstable equidimensional maps, and the number of stable projections of algebraic hypersurfaces

We study the recognition of \(\)-classes of multi-germs in families of corank-1 maps from n-space into n-space. From these recognition conditions we deduce certain geometric properties of bifurcation sets of such families of maps. As applications we give a formula for the number of \(\)-codimension-1 classes of corank-1 multi-germs from ℂn to ℂn and an upper bound for the number of stable projections of algebraic hypersurfaces in ℝn+1 into hyperplanes.

Don’t panic … just do it in parallel!

Parallel coordinates is a methodology for visualizing N-dimensional geometry and multivariate problems. In this self-contained up-to-date overview the aim is to clarify salient points causing difficulties, and point out more sophisticated applications and uses in statistics which are marked by **. Starting from the definition of the parallel-axes multidimensional coordinate system, where a point in Euclidean N-space RN is represented by a polygonal line, it is found that a point ↔ line duality is induced in the Euclidean plane R2. This leads to the development in the projective, P2, rather than the Euclidean plane. Pointers on how to minimize the technical complications and avoid errors are provided. The representation (i.e. visualization) of 1-dimensional objects is obtained from the envelope of the polygonal lines representing the points on their points. On the plane R2 there is a inflection-point ↔ cusp, conies ↔ conies and other potentially useful dualities. A line l ⊂ RN is represented by N − 1 points with a pair of indices in [1, 2, …, N]. This representation also enables the visualization and computation of proximity properties like the minimum distance between pairs of lines [18]. The representation of objects of dimension ≥ 2 is obtained recursively. Specifically, the representation of a p-flat, a plane of dimension 2 ≤ p ≤ N − 1 in RN is obtained from the (p−1)-flats it contains, and which are obtained from the (p−2)-flats and so on all the way down from the points (0-dimensional); hence the recursion. A p-flat is represented by p-points each with (p+1) indices. This is the key message: ** high-dimensional objects may be visualized recursively, in terms of their higher dimensional components, rather than directly from their points. Further, this process is robust so that “near” p-flats are also detected in the same way and very useful tight error bounds are available. The representation of a smooth hypersurface in RN is obtained as the envelope of the tangent hyperplanes. The set of points obtained in this way visually reveal properties like convexity, whether the surface is developable, or ruled. A simpler but ambiguous representation for hypersurfaces is also given together with modeling applications of an algorithm for computing and displaying interior, exterior or surface points.

A fattening principle for fronts propagating by mean curvature plus a driving force

We demonstrate a general situation in which a hypersurface in Rn propagating by mean curvature, plus a nonzero driving force, develops an interior after finite time in the level set formulation of the problem.

Affine Isometric Embedding for Surfaces

A strictly convex hypersurface in Rn can be endowed with a Riemannian metric in a way that is invariant under the group of (equi)affine motions. We study the corresponding isometric embedding problem for surfaces in R3. This problem is formulated in terms of a quasilinear elliptic system of PDE for the Pick form. A negative result is obtained by attempting to invert about the standard embedding of the round sphere as an ellipsoid.

Existence and regularity of weak solutions to the prescribed meancurvature equation for a nonparametric surface

It is known that for the parametric Plateau’s problem, weak solutions can be obtained as critical points of a functional (see [2, 6, 7, 8, 10, 11]). The nonparametric case has been studied for H = H(x,y) (and generally H = H(x1, . . . ,xn) for hypersurfaces in Rn+1) by Gilbarg, Trudinger, Simon, and Serrin, among other authors. It has been proved [5] that there exists a solution for any smooth boundary data if the mean curvature H ′ of ∂ satisfies H ′ ( x1, . . . ,xn )≥ n n−1 ∣∣H (x1, . . . ,xn)∣∣ (1.3)

Separation of hypersurfaces

We give a generalization of results obtained in [15]. LetKn denote the set of embedded hypersurfaces in ℝn+1; for all xeSn and MeKn we denote by CxM the apparent contour ofM in the directionx. Then we give a sufficient condition on W⊂Sn such that the map ϕWKn:Kn →P(T Sn) , defined by ϕWKn (M)=∪{CwM¦ weW}, is injective.

Hypersurfaces with constant scalar curvature in space forms

Let Rn+l(c) be an (n + 1)-dimensional Riemannian manifold with constant sectional curvature c, we also call it space form. When c -1, R"+l(c) = S "+l (i.e. (n + 1)-dimensional unit sphere space); when c = O, Rn+t(c) = E n+l (i.e. (n + 1)-dimensional Euclidean space). Let M be an n-dimensional compact hypersurface in Rn+l(c), and el . . . . . e,, a local orthonormal frame field on M, col . . . . . con its dual coframe field. Then the second fundamental form of M is

Centroaffine minimal hypersurfaces in ? n+1

In this paper the first and the second variation formulas for the area integral of the centroaffine metric of hypersurfaces in ℝ n+1 are calculated, and some interesting examples of stable and unstable centroaffine minimal hypersurfaces are given.

On the existence of surfaces with prescribed mean curvature and boundary in higher dimensions

Given a Lipschitz continuous function H : Rn + 1 → R we construct hypersurfaces in Rn + 1 with prescribed mean curvature H which satisfy a Plateau boundary condition provided the mean curvature function H satisfies a certain isoperimetric condition. For n ≦ 6 these surfaces are free from interior singularities.

Embedded solutions for exterior minimal surface problems

We construct area minimizing hypersurfaces in ℝn+1 which satisfy a Plateau or free boundary condition and become asymptotic to a plane (n≥2) or to the half of a catenoid (n=2). For n≤6 these surfaces are free from singularities.

Hypersurfaces in 𝑅ⁿ whose unit normal has small BMO norm

Let M M be a hypersurface in R d + 1 {{\mathbf {R}}^{d + 1}} whose Gauss map has small BMO norm. This condition is closely related to (but much weaker than) the requirement that the principal curvatures of M M have small L d ( M ) {L^d}\left ( M \right ) norm. (The relationship between these two conditions is a nonlinear geometrical analogue of a classical Sobolev embedding.) This paper deals with the problem of understanding the geometrical constraints imposed on M M by the requirement that the Gauss map have small BMO norm.

A remark on the global embedding problem for three-dimensional CR manifolds

Examples are constructed of compact strictly pseudoconvex real analytic three-dimensional CR manifolds which admit no nonconstant CR functions. Also, the embedding problem for certain CR manifolds is shown to be related to a result of Hadamard on immersed convex hypersurfaces in R n {{\mathbf {R}}^n} .

A generalization of a theorem of Delaunay to rotationalW-hypersurfaces ofσl-type inHn+1andSn+1

In 1841 Delaunay proved that if one rolls a conic section on a line in a plane and then rotates about that line the trace of a focus, one obtains a constant mean curvature surface of revolution in R3. Conversely, all such surfaces, except spheres, are constructed in this way. In 1981, Hsiang and Yu generalized Delaunay's theorem to constant mean curvature rotation hypersurfaces in Rn+1. In 1982, Hsiang further generalized Delaunay's theorem to rotational W-hypersurfaces of σ Γtype in Rn+1. These are hypersurfaces such that the /th-basic symmetric polynomial of the principal curvatures (k^x)), namely,

Oscillatory integrals and Fourier transforms of surface carried measures

We suppose that S S is a smooth hypersurface in R n + 1 {{\mathbf {R}}^{n + 1}} with Gaussian curvature κ \kappa and surface measure d S dS , w w is a compactly supported cut-off function, and we let μ α {\mu _\alpha } be the surface measure with d μ α = w κ α d S d{\mu _\alpha } = w{\kappa ^\alpha }\,dS . In this paper we consider the case where S S is the graph of a suitably convex function, homogeneous of degree d d , and estimate the Fourier transform μ ^ α {\hat \mu _\alpha } . We also show that if S S is convex, with no tangent lines of infinite order, then μ ^ α ( ξ ) {\hat \mu _\alpha }(\xi ) decays as | ξ | − n / 2 |\xi {|^{ - n / 2}} provided α ⩾ [ ( n + 3 ) / 2 ] \alpha \geqslant [(n + 3)/2] . The techniques involved are the estimation of oscillatory integrals; we give applications involving maximal functions.

Integer points on hypersurfaces

Very general hypersurfaces in ℝ4 contain ≪r2+(4/9) integer points in any ball of radiusr>1. As a consequence, an irreducible algebraic hypersurface in ℝn (wheren≥4) which is not a cylinder and is of degreed, contains ≤c(d, n)rn−1−(5/9) integer points in a ball of radiusr. This improves on the known boundc(d, n)rn−(3/2).