Hypersurfaces In Euclidean Space Research Articles

The infinitesimal deformations of hypersurfaces that preserve the Gauss map

Classifying the nonflat hypersurfaces in Euclidean space f : M n → R n + 1 f\colon M^n\to \mathbb {R}^{n+1} that locally admit smooth infinitesimal deformations that preserve the Gauss map infinitesimally was a problem only considered by Schouten in 1928 [Proceedings Amsterdam 31 (1928), pp. 208–218]. He found two conditions that are necessary and sufficient, with the first one being the minimality of the submanifold. The second is a technical condition that does not clarify much about the geometric nature of the hypersurface. In that respect, the parametric solution of the problem given in this note yields that the submanifold has to be Kaehler.

A Characterization of Separable Hypersurfaces in Euclidean Space

A Characterization of Separable Hypersurfaces in Euclidean Space

Geometric characterizations of canal hypersurfaces in Euclidean spaces

In the present paper, firstly we obtain the general expression of canal hypersurfaces in Euclidean n-space and deal with canal hypersurfaces in Euclidean 4-space E4. We compute Gauss map, Gaussian curvature and mean curvature of canal hypersurfaces in E4 and obtain an important relation between the mean and Gaussian curvatures as 3H? = K?3 ? 2. We prove that, the flat canal hypersurfaces in Euclidean 4-space are only circular hypercylinders or circular hypercones and minimal canal hypersurfaces are only generalized catenoids. Also, we state the expression of tubular hypersurfaces in Euclidean spaces and give some results about Weingarten tubular hypersurfaces in E4.

Symmetry of hypersurfaces and the Hopf Lemma

<abstract><p>A classical theorem of A. D. Alexandrov says that a connected compact smooth hypersurface in Euclidean space with constant mean curvature must be a sphere. We give exposition to some results on symmetry properties of hypersurfaces with ordered mean curvature and associated variations of the Hopf Lemma. Some open problems will be discussed.</p></abstract>

Relatives Geometries

In this paper we consider M a fixed hypersurface in Euclidean space and we introduce two types of spaces relative to M, of type I and type II. We observe that when M is a hyperplane, the two geometries coincides with the isotropic geometry. By applying the theory to a Dupin hypersurface M, we define a relative Dupin hypersurface M of type I and type II , we provide necessary and sufficient conditions for a relative hypersurface M to be relative Dupin parameterized by relative lines of curvature, in both spaces. Moreover, we provides a relationship between the Dupin hypersurfaces locally associated to M by a Ribaucour transformation and the type II Dupin hypersurfaces relative M. We provide explicit examples of the Dupin hypersurface relative to a hyperplane, torus, S1 x Rn-1 and S2 x Rn-2, in both spaces.

Generalized Weingarten hypersurfaces of radial support type

In this paper we introduce a class of Weingarten hypersurfaces called generalized Weingarten hypersurfaces of radial support type (in short, RSGW hypersurfaces). These hypersurfaces are parameterized using a new technique consisting in obtain locally any hypersurface in Euclidean space as an envelope of a sphere congruence wherein the other envelope is contained in a sphere. We extend the definition of the classical Appell surfaces to hypersurfaces and we characterize both Appell and RSGW hypersurfaces in terms of a same harmonic function in the sphere. For two-dimensional case, we provide a Weierstrass-type representation for RSGW surfaces and from this we get a Weierstrass-type representation for the classical Appell surfaces. These representations depend on two holomorphic functions in such a way that a same pair of functions provides examples in each class. Using it, we give a classification for the rotational cases for both classes. Furthermore is provided a necessary and sufficient condition on the holomorphic data of these classes so that they are parameterized by lines of curvature. We also prove that a compact, complete RSGW surface is a sphere.

Finding extremals of Lagrangian actions

Given a smooth m-manifold M, a smooth Lagrangian L:TM→R and endpoints x0,xT∈M, we look for an extremal x:[0,T]→M of the action ∫0TL(x(t),ẋ(t))dt satisfying x(0)=x0 and x(T)=xT. When interpolating between endpoints, this amounts to a 2-point boundary value problem for the Euler–Lagrange equation. Single or multiple shooting is one of the most popular methods to solve boundary value problems, but the efficiency of shooting and the quality of solutions depends heavily on initial guesses. In the present paper, by dividing the interval [0,T] into several sub-intervals, on which extremals can be found efficiently by shooting when good initial guesses are available from the geometry of a variational problem, we then adjust all junctions by finding zeros of vector fields associated with the velocities at junctions with Newton’s method. We discuss the cases where L is the difference between kinetic energy and potential, M is a hypersurface in Euclidean space, or M is a Lie group. We make some comparisons in numerical experiments for a double pendulum, for obstacle avoidance by a moving particle on the 2-sphere, and for obstacle avoidance by a planar rigid body.

Convexity estimates for hypersurfaces moving by concave curvature functions

We study fully nonlinear geometric flows that deform strictly k-convex hypersurfaces in Euclidean space with pointwise normal speed given by a concave function of the principal curvatures. Specifically, the speeds we consider are obtained by performing a nonlinear interpolation between the mean and the k-harmonic mean of the principal curvatures. Our main result is a convexity estimate showing that, on compact solutions, regions of high curvature are approximately convex. In contrast to the mean curvature flow, the fully nonlinear flows considered here preserve k-convexity in a Riemannian background, and we show that the convexity estimate carries over to this setting as long as the ambient curvature satisfies a natural pinching condition.

A fractional Michael–Simon Sobolev inequality on convex hypersurfaces

The classical Michael-Simon and Allard inequality is a Sobolev inequality for functions defined on a submanifold of Euclidean space. It is governed by a universal constant independent of the manifold, but displays on the right-hand side an additional $L^p$ term weighted by the mean curvature of the underlying manifold. We prove here a fractional version of this inequality on hypersurfaces of Euclidean space that are boundaries of convex sets. It involves the Gagliardo semi-norm of the function, as well as its $L^p$ norm weighted by the fractional mean curvature of the hypersurface. As an application, we establish a new upper bound for the maximal time of existence in the smooth fractional mean curvature flow of a convex set. The bound depends on the perimeter of the initial set instead of on its diameter.

On the minimality of quasi‐sum production models in microeconomics

Historically, the minimality of surfaces is extremely important in mathematics, and the study of minimal surfaces is a central problem, which has been widely concerned by mathematicians. Meanwhile, the study of the shape and the properties of the production models is a great interest subject in economic analysis. The aim of this paper is to study the minimality of quasi‐sum production functions as graphs in a Euclidean space. We obtain minimal characterizations of quasi‐sum production functions with two or three factors as hypersurfaces in Euclidean spaces. As a result, our results also give a classification of minimal quasi‐sum hypersurfaces in dimensions two and three.

Volume properties and rigidity on self-expanders of mean curvature flow

In this paper, we mainly study immersed self-expander hypersurfaces in Euclidean space whose mean curvatures have some linear growth controls. We discuss the volume growths and the finiteness of the weighted volumes. We prove some theorems that characterize the hyperplanes through the origin as self-expanders. We estimate upper bound of the bottom of the spectrum of the drifted Laplacian. We also give the upper and lower bounds for the bottom of the spectrum of the L-stability operator and discuss the L-stability of some special self-expanders. Besides, we prove that the surfaces \(\Gamma \times \mathbb {R}\) with the product metric are the only complete self-expander surfaces immersed in \(\mathbb {R}^3\) with constant scalar curvature, where \(\Gamma \) is a complete self-expander curve (properly) immersed in \(\mathbb {R}^2\).

The principal curvature theorem and its applications to constant mean curvature hypersurfaces in Euclidean space

The principal curvature theorem and its applications to constant mean curvature hypersurfaces in Euclidean space

Universal Hardy–Sobolev inequalities on hypersurfaces of Euclidean space

In this paper, we study Hardy–Sobolev inequalities on hypersurfaces of [Formula: see text], all of them involving a mean curvature term and having universal constants independent of the hypersurface. We first consider the celebrated Sobolev inequality of Michael–Simon and Allard, in our codimension one framework. Using their ideas, but simplifying their presentations, we give a quick and easy-to-read proof of the inequality. Next, we establish two new Hardy inequalities on hypersurfaces. One of them originates from an application to the regularity theory of stable solutions to semilinear elliptic equations. The other one, which we prove by exploiting a “ground state” substitution, improves the Hardy inequality of Carron. With this same method, we also obtain an improved Hardy or Hardy–Poincaré inequality.

Conformal infinitesimal variations of Euclidean hypersurfaces

In the realm of conformal geometry, we give a parametric classification of the hypersurfaces in Euclidean space that admit nontrivial conformal infinitesimal variations. A parametric classification of the Euclidean hypersurfaces that allow a nontrivial conformal variation was obtained by E. Cartan in 1917. In particular, we show that the class of hypersurfaces studied here is much larger than the one characterized by Cartan.

An extension of Jellett-Liebmann theorem to hypersurfaces in Euclidean space

An extension of Jellett-Liebmann theorem to hypersurfaces in Euclidean space

Classifications of quotion Yamabe solitons on Euclidean submanifolds with concurrent vector fields

The aim of this paper is to obtain some results for quotion Yamabe solitons with concurrent vector fields. We prove quotion Yamabe soliton [Formula: see text] on a hypersurface in Euclidean space [Formula: see text] contained either in a hyperplane or in a sphere [Formula: see text].

Classification of generalized Yamabe solitons in Euclidean spaces

In this paper, we consider generalized Yamabe solitons which include many notions, such as Yamabe solitons, almost Yamabe solitons, [Formula: see text]-almost Yamabe solitons, gradient [Formula: see text]-Yamabe solitons and conformal gradient solitons. We completely classify the generalized Yamabe solitons on hypersurfaces in Euclidean spaces arisen from the position vector field.

Rotational λ – hypersurfaces in Euclidean spaces

Self-similar flows arise as special solution of the mean curvature flow that preserves the shape of the evolving submanifold. In addition, \lambda -hypersurfaces are the generalization of self-similar hypersurfaces. In the present article we consider \lambda -hypersurfaces in Euclidean spaces which are the generalization of self-shrinkers. We obtained some results related with rotational hypersurfaces in Euclidean 4-space \mathbb{R}^{4} to become self-shrinkers. Furthermore, we classify the general rotational \lambda -hypersurfaces with constant mean curvature. As an application, we give some examples of self-shrinkers and rotational \lambda -hypersurfaces in \mathbb{R}^{4}.

Volume preserving flow by powers of the $k$th mean curvature

We consider the flow of closed convex hypersurfaces in Euclidean space $\mathbb{R}^{n+1}$ with speed given by a power of the $k$‑th mean curvature $E_k$ plus a global term chosen to impose a constraint involving the enclosed volume $V_{n+1}$ and the mixed volume $V_{n+1-k}$ of the evolving hypersurface. We prove that if the initial hypersurface is strictly convex, then the solution of the flow exists for all time and converges to a round sphere smoothly. No curvature pinching assumption is required on the initial hypersurface.

Complete constant mean curvature hypersurfaces in Euclidean space of dimension four or higher

In this article we provide a general construction when $n\ge3$ for immersed in Euclidean $(n+1)$-space, complete, smooth, constant mean curvature hypersurfaces of finite topological type (in short CMC $n$-hypersurfaces). More precisely our construction converts certain graphs in Euclidean $(n+1)$-space to CMC $n$-hypersurfaces with asymptotically Delaunay ends in two steps: First appropriate small perturbations of the given graph have their vertices replaced by round spherical regions and their edges and rays by Delaunay pieces so that a family of initial smooth hypersurfaces is constructed. One of the initial hypersurfaces is then perturbed to produce the desired CMC $n$-hypersurface which depends on the given family of perturbations of the graph and a small in absolute value parameter $\underline\tau$. This construction is very general because of the abundance of graphs which satisfy the required conditions and because it does not rely on symmetry requirements. For any given $k\ge2$ and $n\ge3$ it allows us to realize infinitely many topological types as CMC $n$-hypersurfaces in $\mathbb R^{n+1}$ with $k$ ends. Moreover for each case there is a plethora of examples reflecting the abundance of the available graphs. This is in sharp contrast with the known examples which in the best of our knowledge are all (generalized) cylindrical obtained by ODE methods and are compact or with two ends. Furthermore we construct embedded examples when $k\ge3$ where the number of possible topological types for each $k$ is finite but tends to $\infty$ as $k\to\infty$.