Hypersurfaces In Rn Research Articles

Inequalities for some maximal functions. II

Let S S be a smooth compact hypersurface in R n {{\mathbf {R}}^n} , and let μ \mu be a measure on S S , absolutely continuous with respect to surface measure. For t t in R + , μ t {{\mathbf {R}}^ + },{\mu _t} denotes the dilate of μ \mu by t t , normalised to have the same total variation as μ \mu : for f f in S ( R n ) , μ # f \mathcal {S}({{\mathbf {R}}^n}),{\mu ^\# }f denotes the maximal function sup t > 0 | μ t ∗ f | {\sup _{t > 0}}|{\mu _t}\ast f| . We seek conditions on μ \mu which guarantee that the a priori estimate \[ ‖ μ # f ‖ p ≤ C ‖ f ‖ p , f ∈ S ( R n ) , \left \| \mu ^\# f\right \|_p \leq C\left \| f \right \|_p, \quad f \in S(\mathbf {R}^n), \] holds; this estimate entails that the sublinear operator μ # {\mu ^\# } extends to a bounded operator on the Lebesgue space L p ( R n ) {L^p}({{\mathbf {R}}^n}) . Our methods generalise E. M. Stein’s treatment of the "spherical maximal function" [5]: a study of "Riesz operators", g g -functions, and analytic families of measures reduces the problem to that of obtaining decay estimates for the Fourier transform of μ \mu . These depend on the geometry of S S and the relation between μ \mu and surface measure on S S . In particular, we find that there are natural geometric maximal operators limited on L p ( R n ) {L^p}({{\mathbf {R}}^n}) if and only if p ∈ ( q , ∞ ] ; q p \in (q,\infty ];q is some number in ( 1 , ∞ ) (1,\infty ) , and may be greater than 2 2 . This answers a question of S. Wainger posed by Stein [6]>.

Integer points on curves and surfaces

Various upper bounds are given for the number of integer points on plane curves, on surfaces and hypersurfaces. We begin with a certain class of convex curves, we treat rather general surfaces in ℝ3 which include algebraic surfaces with the exception of cylinders, and we go on to hypersurfaces in ℝn with nonvanishing Gaussian curvature.

Minimal surfaces with isolated singularities

For n≥3, there exists an embedded minimal hypersurface in Rn+1 which has an isolated singularity but which is not a cone. Each example constructed here is asymptotic to a given, completely arbitrary, nonplanar minimal cone and is stable in case the cone satisfies a strict stability inequality.

The duals of generic space curves and complete intersections

In a previous paper we discussed the duals of generic hypersurfaces: both smooth hypersurfaces in ℝn and algebraic hypersurfaces in real or complex projective space ℙn. In this note we show how to extend the methods of [1] to cover the case of complete intersections in ℙn and preface this with a brief discussion on the contact of space curves in ℝn with planes. We shall use the notation of [1].

On generic hypersurfaces in ℝn

In this paper we consider certain questions concerning the differential geometry of generic hypersurfaces in ℝn. Our results prove, for example, that the curve of rib points of a generic surface in ℝ3 has transverse self-intersections.In (4) Porteous discussed (amongst other things) the generic geometry of curves and surfaces in ℝ3. Subsequently Looijenga ((3) and see also (5)) gave a more precise definition of the term generic and showed that an open dense subset of smooth embeddings of manifolds in Euclidean space were indeed generic.

Sufficient criteria of convexity

In this article a survey of criteria of convexity of sets and hypersurfaces inRn, in Riemannian and infinite-dimensional linear spaces, is made. Included are the following sections: support property; local convexity; local support; curvature of the boundary; section by planes; uniqueness of closest points; systems of sets.

Hypersurfaces of nonnegative curvature in a Hilbert space

In this paper we prove the following generalizations of known theorems about hypersurfaces in R n {{\mathbf {R}}^n} : Let M M be a hypersurface in a Hilbert space. (1) If on M M the sectional curvature K ( σ ) K(\sigma ) is nonnegative for every 2 2 -plane section σ \sigma and if K ( σ ) > 0 K(\sigma ) > 0 for at least one σ \sigma , then M M is the boundary of a convex body. (2) If K ( σ ) = 0 K(\sigma ) = 0 for all σ \sigma , then M M is a hypercylinder. The main tool in these theorems is Smale’s infinite dimensional Sard’s theorem.

Orientability of hypersurfaces in 𝑅ⁿ

We give an elementary proof for the fact that a hypersurface of codimension 1 in Rn is always orientable [a surface of the type we consider is, by definition, a closed, not necessarily compact, subset of Rn that near each of its points is the set of zeros of some realvalued Coo-function with nonzero gradient]. Another way to say this: a differentiable manifold (without boundary) of dimension n-1 cannot be Cco-embedded as subset of Rn; here closed is needed as the Mobius strip (without boundary) in R3 shows. This is usually proved (for compact manifolds, without any differentiability hypothesis) by invoking Alexander duality [2]. Suppose we had such a hypersurface M in Rn. We take nonorientable to mean: there exists a loop (closed curve) in M such that the normal to M, when transported around the loop in a continuous fashion, comes back with the opposite direction. By considering a point on the normal a small distance off M, moving it arotund the loop and then connecting along the normal from one side of M to the other, we manufacture a C0-curve y in Rn that meets M in exactly one point, and is transversal to M at this point [the tangent to y is not tangent to M]. y can be contracted to a point in Rn; this amounts to a Coo-map f of the unit disk D2 into Rn that on the boundary S of D2 yields y. We now make f transversal to M, i.e., we bring it into position relative to M, so that f(D2) is not tangent to M at any common point; this can be done without changing f on SI, since f is already transversal there (see [1], [3], [4], [5]). [The general transversality theorem looks formidable, but is really quite simple, particularly in the case of codimension 1; one modifies f, locally, by adding suitable affine-linear functions, multiplied by cutoff functions; in the analogous piecewise linear case one shifts the images of the vertices of some triangulation of D2 into general position. ]