Hypersurfaces In Rn Research Articles

A note on new weighted geometric inequalities for hypersurfaces in ℝⁿ

In this note, we prove a family of sharp weighed inequalities which involve weighted k k -th mean curvature integral and two distinct quermassintegrals for closed hypersurfaces in R n \mathbb {R}^n . This inequality generalizes the corresponding result of Wei and Zhou [Bull. Lond. Math. Soc. 55 (2023), pp. 263–281] where their proof is based on earlier results of Kwong-Miao [Pacific J. Math. 267 (2014), pp. 417–422; Commun. Contemp. Math. 17 (2015), p. 1550014]. Here we present a proof which does not rely on Kwong-Miao’s results.

Using Lie Sphere Geometry to Study Dupin Hypersurfaces in Rn

Using Lie Sphere Geometry to Study Dupin Hypersurfaces in Rn

Weighted Alexandrov–Fenchel type inequalities for hypersurfaces in Rn$\mathbb {R}^n$

AbstractIn this paper, we prove the following geometric inequalities in the Euclidean space , which are weighted Alexandrov–Fenchel type inequalities, provided that is a star‐shaped and ‐convex hypersurface. Equality holds if and only if is a coordinate sphere in . As an application, by letting in the above inequality, we obtain a lower bound for the outer radius in terms of the curvature integrals for star‐shaped and ‐convex hypersurfaces.

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The classical Cauchy surface area formula states that the surface area of the boundary ∂K=Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\partial K=\\Sigma $\\end{document} of any n-dimensional convex body in the n-dimensional Euclidean space Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{R}^{n}$\\end{document} can be obtained by the average of the projected areas of Σ along all directions in Sn−1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{S}^{n-1}$\\end{document}. In this note, we generalize the formula to the boundary of arbitrary n-dimensional submanifold in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{R}^{n}$\\end{document} by introducing a natural notion of projected areas along any direction in Sn−1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{S}^{n-1}$\\end{document}. This surface area formula derived from the new notion coincides with not only the result of the Crofton formula but also with that of De Jong (Math. Semesterber. 60(1):81–83, 2013) by using a tubular neighborhood. We also define the projected r-volumes of Σ onto any r-dimensional subspaces and obtain a recursive formula for mean projected r-volumes of Σ.

Non-homogeneous fully nonlinear contracting flows of convex hypersurfaces

Abstract We consider a general class of non-homogeneous contracting flows of convex hypersurfaces in R n + 1 ${\mathbb{R}}^{n+1}$ , and prove the existence and regularity of the flow before extincting to a point in finite time.

Flow by 𝜎_{𝑘} curvature to the Orlicz Christoffel-Minkowski problem

In this paper, we consider the anisotropic curvature flow of smooth, origin-symmetric, uniformly convex hypersurfaces in R n + 1 \mathbb {R}^{n+1} . The flow exists for all time and converges smoothly to a solution of the even Orlicz Christoffel-Minkowski problem. Our proof also gives an approach to the solution of the L p L_p Christoffel-Minkowski problem.

Topology of real multi-affine hypersurfaces and a homological stability property

Let R be a real closed field. We prove that the number of semi-algebraically connected components of a real hypersurface in Rn defined by a multi-affine polynomial of degree d is bounded by 2d−1. This bound is sharp and is independent of n (as opposed to the classical bound of d(2d−1)n−1 on the Betti numbers of hypersurfaces defined by arbitrary polynomials of degree d in Rn due to Petrovskiĭ and Oleĭnik, Thom and Milnor). Moreover, we show there exists c>1, such that given a sequence (Bn)n>0 where Bn is a closed ball in Rn of positive radius, there exist hypersurfaces (Vn⊂Rn)n>0 defined by symmetric multi-affine polynomials of degree 4, such that ∑i⩽5bi(Vn∩Bn)>cn, where bi(⋅) denotes the i-th Betti number with rational coefficients. Finally, as an application of the main result of the paper we verify a representational stability conjecture due to Basu and Riener on the cohomology modules of symmetric real algebraic sets for a new and much larger class of symmetric real algebraic sets than known before.

Entropy bounds, compactness and finiteness theorems for embedded self-shrinkers with rotational symmetry

Abstract In this work, we study the space of complete embedded rotationally symmetric self-shrinking hypersurfaces in ℝ n + 1 {\mathbb{R}^{n+1}} . First, using comparison geometry in the context of metric geometry, we derive explicit upper bounds for the entropy of all such self-shrinkers. Second, as an application we prove a smooth compactness theorem on the space of all such shrinkers. We also prove that there are only finitely many such self-shrinkers with an extra reflection symmetry.

Classification of isotropic Calabi hypersurfaces

The classifications of locally strongly convex isotropic equiaffine spheres and isotropic centroaffine hypersurfaces have been completed in the last decade, see [1] and [4]. In this paper, we consider isotropic Calabi hypersurfaces (also called graph immersion with Calabi normalization) and obtain a complete classification in the affine space Rn+1. As a corollary, we give a new characterization of elliptic paraboloid: Elliptic paraboloids are the only constant isotropic Calabi hypersurfaces in Rn+1.

Reconstruction of polytopes from the modulus of the Fourier transform with small wave length

Abstract Let 𝒫 {\mathcal{P}} be an n-dimensional convex polytope and let 𝒮 {\mathcal{S}} be a hypersurface in ℝ n {\mathbb{R}^{n}} . This paper investigates potentials to reconstruct 𝒫 {\mathcal{P}} , or at least to compute significant properties of 𝒫 {\mathcal{P}} , if the modulus of the Fourier transform of 𝒫 {\mathcal{P}} on 𝒮 {\mathcal{S}} with wave length λ, i.e., | ∫ 𝒫 e - i ⁢ 1 λ ⁢ 𝐬 ⋅ 𝐱 ⁢ 𝐝𝐱 | for ⁢ 𝐬 ∈ 𝒮 , \biggl{\lvert}\int_{\mathcal{P}}e^{-i\frac{1}{\lambda}\mathbf{s}\cdot\mathbf{x% }}\,\mathbf{dx}\biggr{\rvert}\quad\text{for }\mathbf{s}\in\mathcal{S}, is given, λ is sufficiently small and 𝒫 {\mathcal{P}} and 𝒮 {\mathcal{S}} have some well-defined properties. The main tool is an asymptotic formula for the Fourier transform of 𝒫 {\mathcal{P}} with wave length λ when λ → 0 {\lambda\rightarrow 0} . The theory of X-ray scattering of nanoparticles motivates this study, since the modulus of the Fourier transform of the reflected beam wave vectors is approximately measurable on a half sphere in experiments.

Convexity estimate for translating solitons of concave fully nonlinear extrinsic geometric flows in [formula omitted

The main result of this paper is a convexity estimate for translating solitons of extrinsic geometric flows which evolve under a 1-homogeneous concave function in the principal curvatures. In addition, we show examples of these hypersurfaces in Rn+1 for particular functions.

Asymptotic convergence for a class of anisotropic curvature flows

In this paper, by using new auxiliary functions, we study a class of contracting flows of closed, star-shaped hypersurfaces in Rn+1 with speed rαβσk1β, where σk is the k-th elementary symmetric polynomial of the principal curvatures, α, β are positive constants and r is the distance from points on the hypersurface to the origin. We obtain convergence results under some assumptions of k, α, β. When k≥2, 0<β≤1, α≥β+k, we prove that the k-convex solution to the flow exists for all time and converges smoothly to a sphere after normalization, in particular, we generalize Li-Sheng-Wang's result from uniformly convex to k-convex. When k≥2, β=k, α≥2k, we prove that the k-convex solution to the flow exists for all time and converges smoothly to a sphere after normalization, in particular, we generalize Ling Xiao's result from k=2 to k≥2.

Singular rank one perturbations

In this paper, A = B + V represents a self-adjoint operator acting on a Hilbert space H. We set a general theoretical framework and obtain several results for singular perturbations of A of the type Aβ = A + βτ*τ for τ being a functional defined in a subspace of H. In particular, we apply these results to Hβ = −Δ + V + β|δ⟩⟨δ|, where δ is the singular perturbation given by δ(φ) = ∫Sφ dσ, where S is a suitable hypersurface in Rn. Using the fact that the singular perturbation τ*τ is a sort of rank one perturbation of the operator A, it is possible to prove the invariance of the essential spectrum of A under these singular perturbations. The main idea is to apply an adequate Krein’s formula in this singular framework. As an additional result, we found the corresponding relationship between the Green’s functions associated with the operators H0 = Δ + V and Hβ, and we give a result about the existence of a pure point spectrum (eigenvalues) of Hβ. We also study the case β goes to infinity.

A fully nonlinear locally constrained anisotropic curvature flow

Given a smooth positive function F∈C∞(Sn) such that the square of its positive 1-homogeneous extension on Rn+1∖{0} is uniformly convex, the Wulff shape WF is a smooth uniformly convex body in the Euclidean space Rn+1 with F being the support function of the boundary ∂WF. In this paper, we introduce the fully nonlinear locally constrained anisotropic curvature flow ∂∂tX=(1−Ek1/kσF)νF,k=2,…,nin the Euclidean space, where Ek denotes the normalized kth anisotropic mean curvature with respect to the Wulff shape WF, σF the anisotropic support function and νF the outward anisotropic unit normal of the evolving hypersurface. We show that starting from a smooth, closed and strictly convex hypersurface in Rn+1 (n≥2), the smooth solution of the flow exists for all positive time and converges smoothly and exponentially to a scaled Wulff shape. A nice feature of this flow is that it improves a certain isoperimetric ratio. Therefore by the smooth convergence of the above flow, we provide a new proof of a class of the Alexandrov–Fenchel inequalities for anisotropic mixed volumes of smooth convex domains in the Euclidean space.

Self-similar solutions to fully nonlinear curvature flows by high powers of curvature

Abstract In this paper, we investigate closed strictly convex hypersurfaces in ℝ n + 1 {\mathbb{R}^{n+1}} which shrink self-similarly under a large family of fully nonlinear curvature flows by high powers of curvature. When the speed function is given by powers of a homogeneous of degree 1 and inverse concave function of the principal curvatures with power greater than 1, we prove that the only such hypersurfaces are round spheres. We also prove that slices are the only closed strictly convex self-similar solutions to such curvature flows in the hemisphere 𝕊 + n + 1 {\mathbb{S}^{n+1}_{+}} with power greater than or equal to 1.

On the geometry of tangent bundle of a hypersurface in Rn+1

In this paper, tangent bundle $TM$ of the hypersurface $M$ in $% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{n+1}$ has been studied. For hypersurface $M$ given by immersion $f:M\rightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{n+1},$ considering the fact that $F=df:TM\rightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{2n+2}$ is also immersion, $TM$ is treated as a submanifold of $% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{2n+2}.$ Firstly, an induced metric which is called rescaled induced metric has been defined on $TM,$ and the Levi-Civita connection has been calculated for this metric. Next, curvature tensors of tangent bundle $TM$ have been obtained. Finally, the orthonormal frame at the point $(p,u)\in TM$ has been defined and some curvature properties of such a tangent bundle by means of orthonormal frame for a given point have been investigated.

On D-invariant points and local Taylor interpolation on algebraic hypersurfaces in RN

We give the definition of D-invariant points on an irreducible algebraic hypersurface V in RN. We show that every regular point on irreducible quadratic hypersurface in RN is D-invariant. We prove that the local Taylor interpolation projector at a regular point a 2 V is an ideal projector if and only if a is D-invariant.

The global geometry of surfaces with prescribed mean curvature in ℝ³

We develop a global theory for complete hypersurfaces inRn+1\mathbb {R}^{n+1}whose mean curvature is given as a prescribed function of its Gauss map. This theory extends the usual one of constant mean curvature hypersurfaces inRn+1\mathbb {R}^{n+1}, and also that of self-translating solitons of the mean curvature flow. For the particular casen=2n=2, we will obtain results regarding a priori height and curvature estimates, non-existence of complete stable surfaces, and classification of properly embedded surfaces with at most one end.

A Bernstein type theorem for minimal hypersurfaces via Gauss maps

Let M be an n-dimensional smooth oriented complete embedded minimal hypersurface in Rn+1 with Euclidean volume growth. We show that if the image under the Gauss map of M avoids some neighborhood of a half-equator, then M must be an affine hyperplane.

The rigidity and stability of complete 𝑓-minimal hypersurfaces in ℝⁿ×𝕊¹(𝕒)

In this paper we consider f f -minimal hypersurfaces in the product space R n × S 1 ( a ) \mathbb {R}^n \times \mathbb {S}^1(a) where ( R n , e − f d μ ) (\mathbb {R}^n,e^{-f}d\mu ) is the standard Gaussian space with d μ d\mu being the standard volume element on R n \mathbb {R}^n . By introducing a globally defined smooth function α \alpha , we shall derive some interesting differential identities by which we are able to prove several rigidity theorems. We also study the stability properties of some standard examples. As a result, we prove that R n × { s 0 } ↪ R n × S 1 ( a ) \mathbb {R}^n \times \{s_0\} \hookrightarrow \mathbb {R}^n \times \mathbb {S}^1(a) is the only complete, proper, and stable f f -minimal hypersurface.