Hypersurfaces Of Revolution Research Articles

Critical Lengths of Steklov Eigenvalues of Hypersurfaces of Revolution in Euclidean Space

We study the Steklov problem on hypersurfaces of revolution with two boundary components in Euclidean space and focus on the phenomenon of critical length, at which a Steklov eigenvalue is maximized. In this article, we conjecture that, in any dimension, there is a finite number of infinite critical length. To investigate this, we develop an algorithm to efficiently perform numerical experiments, providing support to our conjecture. Furthermore, we prove the conjecture in dimension n = 3 and n = 4.

Sharp upper bounds for Steklov eigenvalues of a hypersurface of revolution with two boundary components in Euclidean space

R'esum'eWe investigate the question of sharp upper bounds for the Steklov eigenvalues of a hypersurface of revolution in Euclidean space with two boundary components, each isometric to $${\mathbb {S}}^{n-1}$$ S n - 1 . For the case of the first non zero Steklov eigenvalue, we give a sharp upper bound $$B_n(L)$$ B n ( L ) (that depends only on the dimension $$n \ge 3$$ n ≥ 3 and the meridian length $$L>0$$ L > 0 ) which is reached by a degenerated metric $$g^*$$ g ∗ that we compute explicitly. We also give a sharp upper bound $$B_n$$ B n which depends only on n. Our method also permits us to prove some stability properties of these upper bounds.

A Hypersurfaces of Revolution Family in the Five-Dimensional Pseudo-Euclidean Space E25

We present a family of hypersurfaces of revolution distinguished by four parameters in the five-dimensional pseudo-Euclidean space E25. The matrices corresponding to the fundamental form, Gauss map, and shape operator of this family are computed. By utilizing the Cayley–Hamilton theorem, we determine the curvatures of the specific family. Furthermore, we establish the criteria for maximality within this framework. Additionally, we reveal the relationship between the Laplace–Beltrami operator of the family and a 5×5 matrix.

Hypersurfaces of revolution family supplying $ \Delta \mathfrak{r} = \mathcal{A}\mathfrak{r} $ in pseudo-Euclidean space $ \mathbb{E}_{3}^{7} $

<abstract><p>In this study, we introduce a family of hypersurfaces of revolution characterized by six parameters in the seven-dimensional pseudo-Euclidean space $ {\mathbb{E}}_{3}^{7} $. These hypersurfaces exhibit intriguing geometric properties, and our aim is to analyze them in detail. To begin, we calculate the matrices corresponding to the fundamental form, Gauss map, and shape operator associated with this hypersurface family. These matrices provide essential information about the local geometry of the hypersurfaces, including their curvatures and tangent spaces. Using the Cayley-Hamilton theorem, we employ matrix algebra techniques to determine the curvatures of the hypersurfaces. This theorem allows us to express the characteristic polynomial of a matrix in terms of the matrix itself, enabling us to compute the curvatures effectively. In addition, we establish equations that describe the interrelation between the mean curvature and the Gauss-Kronecker curvature of the hypersurface family. These equations provide insights into the geometric behavior of the surfaces and offer a deeper understanding of their intrinsic properties. Furthermore, we investigate the relationship between the Laplace-Beltrami operator, a differential operator that characterizes the geometry of the hypersurfaces, and a specific $ 7\times 7 $ matrix denoted as $ \mathcal{A} $. By studying this relation, we gain further insights into the geometric structure and differential properties of the hypersurface family. Overall, our study contributes to the understanding of hypersurfaces of revolution in $ {\mathbb{ E}}_{3}^{7} $, offering mathematical insights and establishing connections between various geometric quantities and operators associated with this family.</p></abstract>

Sharp bounds for Steklov eigenvalues on star-shaped domains

In this article, we consider Steklov eigenvalue problem on star-shaped bounded domain Ω in hypersurface of revolution and paraboloid, P = {(x, y, z) ∈ ℝ3 : z = x2 + y2}. A sharp lower bound is derived for all Steklov eigenvalues of Ω in terms of the Steklov eigenvalues of the largest geodesic ball contained in Ω with the same center as Ω. This work is a generalization of a result given by Kuttler and Sigillito (SIAM Rev 10:368 − 370, 1968) on a star-shaped bounded domain in ℝ2.

Constructing 3D Self-Supporting Surfaces with Isotropic Stress Using 4D Minimal Hypersurfaces of Revolution

This article presents a new computational framework for constructing 3D self-supporting surfaces with isotropic stress. Inspired by the self-supporting property of catenary and the fact that catenoid (the surface of revolution of the catenary curve) is a minimal surface, we discover the relation between 3D self-supporting surfaces and 4D minimal hypersurfaces (which are 3-manifolds). Lifting the problem into 4D allows us to convert gravitational forces into tensions and reformulate the equilibrium problem to total potential energy minimization, which can be solved using a variational method. We prove that the hyper-generatrix of a 4D minimal hyper-surface of revolution is a 3D self-supporting surface, implying that constructing a 3D self-supporting surface is equivalent to volume minimization. We show that the energy functional is simply the surface’s gravitational potential energy, which in turn can be converted into a surface reconstruction problem with mean curvature constraint. Armed with our theoretical findings, we develop an iterative algorithm to construct 3D self-supporting surfaces from triangle meshes. Our method guarantees convergence and can produce near-regular triangle meshes, thanks to a local mesh refinement strategy similar to centroidal Voronoi tessellation. It also allows users to tune the geometry via specifying either the zero potential surface or its desired volume. We also develop a finite element method to verify the equilibrium condition on 3D triangle meshes. The existing thrust network analysis methods discretize both geometry and material by approximating the continuous stress field through uniaxial singular stresses, making them an ideal tool for analysis and design of beam structures. In contrast, our method works on piecewise linear surfaces with continuous material. Moreover, our method does not require the 3D-to-2D projection, therefore it also works for both height and non-height fields.

Steklov Eigenvalues of Submanifolds with Prescribed Boundary in Euclidean Space

We obtain upper and lower bounds for Steklov eigenvalues of submanifolds with prescribed boundary in Euclidean space. A very general upper bound is proved, which depends only on the geometry of the fixed boundary and on the measure of the interior. Sharp lower bounds are given for hypersurfaces of revolution with connected boundary: We prove that each eigenvalue is uniquely minimized by the ball. We also observe that each surface of revolution with connected boundary is Steklov isospectral to the disk.

Rotational hypersurfaces with $L_r$-pointwise 1-type Gauss map

In this paper, we study hypersurfaces in $\E^{n+1}$ which Gauss map $G$ satisfies the equation $L_rG = f(G + C)$ for a smooth function $f$ and a constant vector $C$, where $L_r$ is the linearized operator of the $(r + 1)$th mean curvature of the hypersurface, i.e., $L_r(f)=tr(P_r\circ\nabla^2f)$ for $f\in \mathcal{C}^\infty(M)$, where $P_r$ is the $r$th Newton transformation, $\nabla^2f$ is the Hessian of $f$, $L_rG=(L_rG_1,\ldots,L_rG_{n+1}), G=(G_1,\ldots,G_{n+1})$. We show that a rational hypersurface of revolution in a Euclidean space $\E^{n+1}$ has $L_r$-pointwise 1-type Gauss map of the second kind if and only if it is a right n-cone.

Loxodromes on hypersurfaces of revolution

Loxodromes on hypersurfaces of revolution

Linear and bilinear restriction to certain rotationally symmetric hypersurfaces

Conditional on Fourier restriction estimates for elliptic hypersurfaces, we prove optimal restriction estimates for polynomial hypersurfaces of revolution for which the defining polynomial has non-negative coefficients. In particular, we obtain uniform—depending only on the dimension and polynomial degree—estimates for restriction with affine surface measure, slightly beyond the bilinear range. The main step in the proof of our linear result is an (unconditional) bilinear adjoint restriction estimate for pieces at different scales.

Radial multipliers and restriction to surfaces of the Fourier transform in mixed-norm spaces

In this article we revisit some classical conjectures in harmonic analysis in the setting of mixed norm spaces \(L_{rad}^{p}L_{ang}^{2}\left( \mathbb {R}^{n}\right) \). We produce sharp bounds for the restriction of the Fourier transform to compact hypersurfaces of revolution in the mixed norm setting and study an extension of the disc multiplier. We also present some results for the discrete restriction conjecture and state an intriguing open problem.

Gabriel's Horn: A Revolutionary Tale

SummaryWe generalize hypersurfaces of revolution by allowing the profile curve to be dependent on more than one variable. We call these more general objects spherical arrays and we use them to introduce the reader to volume calculations, which are normally reserved for a more advanced course in differential geometry. The rich symmetry of the spherical arrays allow us to build objects with properties reminiscent of those for which Gabriel's Horn became a cause célèbre.

Characterizing Power Functions by Hypervolumes of Revolution

SummaryA power function is characterized by a certain constant volume ratio associated with the surface of revolution generated by the graph of the function. We generalize this characterization to include hypersurfaces of revolution and find that power functions are similarly identified by the analogous ratio of hypervolumes of revolution. We write this ratio as an explicit function of the exponent of the power function and the dimension of the hypersurface.

Self-shrinkers with a rotational symmetry

In this paper we present a new family of non-compact properly embedded, self-shrinking, asymptotically conical, positive mean curvature ends $\Sigma^n\subseteq\mathbb{R}^{n+1}$ that are hypersurfaces of revolution with circular boundaries. These hypersurface families interpolate between the plane and half-cylinder in $\mathbb{R}^{n+1}$, and any rotationally symmetric self-shrinking non-compact end belongs to our family. The proofs involve the global analysis of a cubic-derivative quasi-linear ODE. We also prove the following classification result: a given complete, embedded, self-shrinking hypersurface of revolution $\Sigma^n$ is either a hyperplane $\mathbb{R}^{n}$, the round cylinder $\mathbb{R}\times S^{n-1}$ of radius $\sqrt{2(n-1)}$, the round sphere $S^n$ of radius $\sqrt{2n}$, or is diffeomorphic to an $S^1\times S^{n-1}$ (i.e. a "doughnut" as in [Ang], which when $n=2$ is a torus). In particular for self-shrinkers there is no direct analogue of the Delaunay unduloid family. The proof of the classification uses translation and rotation of pieces, replacing the method of moving planes in the absence of isometries.

Hypersurfaces of revolution with proportional principal curvatures

Abstract We classify hypersurfaces of revolution for which the principal curvatures are proportional. These hypersurfaces form a one-parameter family indexed by the reals. The subsequent classification unites at least two important families of hypersurfaces of revolution: the generalized catenoids and the equizonal ovaloids.

Stability of constant mean curvature hypersurfaces of revolution in hyperbolic space

In this article, by solving a nonlinear differential equation, we prove the existence of a one parameter family of constant mean curvature hypersurfaces in the hyperbolic space with two ends. Then, we study the stability of these hypersurfaces.

On the smoothness of the equizonal ovaloids

We show that the n-dimensional equizonal ovaloids are analytic when n is even and are of exactly Cn-1 smoothness when n is odd. This substantially improves the previously published result on the smoothness of the even-dimensional equizonal ovaloids and slightly corrects the previously published statement regarding the smoothness of the odd-dimensional equizonal ovaloids. Our methods should be generally useful in determining the degree of smoothness of surfaces and hypersurfaces of revolution generated by piecewise-defined profile curves. In particular, they include a novel and elegant application of Bernstein’s theory of absolutely monotonic functions.

Ricci solitons on locally conformally flat hypersurfaces in space forms

We study Ricci solitons on locally conformally flat hypersurfaces Mn in space forms M˜n+1(c) of constant sectional curvature c with potential vector field a principal curvature eigenvector of multiplicity one. We show that in Euclidean space, Mn is a hypersurface of revolution given in terms of a solution of some non-linear ODE. Hence there exists infinitely many mutually non-congruent Ricci solitons of this type. Furthermore when c≥0 and Mn is complete, the Ricci soliton is gradient and in the case it is shrinking, Mn must be the product of the real line and the (n−1)-sphere.

Symmetry-Breaking for Immersed Constant Mean Curvature Hypersurfaces

Abstract In this paper we prove the existence of constant mean curvature hypersurfaces which are cylindrically bounded and which bifurcate from the family of immersed constant mean curvature hypersurface of revolution. Based on the study of the spectrum of the Jacobi operator (the linearized mean curvature) about this family, the existence of new branches follows from a bifurcation result of Crandall and Rabinowitz.

Central Cross-Sections Make Surfaces of Revolution Quadric

We prove here that when all planes transverse and nearly perpendicular to the axis of a surface of revolution intersect it in loops having central symmetry, the surface must be quadric. It follows that the quadrics are the only surfaces of revolution without skewloops. Similar statements hold for hypersurfaces of revolution in higher dimensions.