Hypersurfaces In Euclidean Space Research Articles

Higher-dimensional linking integrals

We derive an integral formula for the linking number of two submanifolds of the $n$-sphere $S^n$, of the product $S^n \times \mathbb {R}^m$, and of other manifolds which appear as “nice” hypersurfaces in Euclidean space. The formulas are geometrically meaningful in that they are invariant under the action of the special orthogonal group on the ambient space.

A note on strict ℂ-convexity

We establish a relation between strict ℂ-convexity of a real hypersurface of ℂn and the behavior of its complex Gauss map. In that way we recover—with an improvement on the regularity—the known results about the topology of these hypersurfaces by using elementary differential geometric arguments. Our approach can be though of as being a complex analog of the description of strictly convex hypersurfaces in Euclidean space via Morse functions associated to pencils of hyperplanes.

A gluing construction for prescribed mean curvature

The gluing technique is used to construct hypersurfaces in Euclidean space having approximately constant prescribed mean curvature. These surfaces are perturbations of unions of finitely many spheres of the same radius assembled end-to-end along a line segment. The condition on the existence of these hypersurfaces is the vanishing of the sum of certain integral moments of the spheres with respect to the prescribed mean curvature function.

A geometric perspective on the generalized Cobb–Douglas production functions

In this work we obtain an interesting link between some fundamental notions in the theory of production functions and the differential geometry of hypersurfaces in Euclidean spaces. In fact we establish that a generalized Cobb–Douglas production function has decreasing/increasing return to scale if and only if the corresponding hypersurface has positive/negative Gaussian curvature. Moreover, this production function has constant return to scale if and only if the corresponding hypersurface is developable.

Two-ended $r$-minimal hypersurfaces in Euclidean space

Two-ended $r$-minimal hypersurfaces in Euclidean space

Contact properties of codimension 2 submanifolds with flat normal bundle

Given an immersed submanifold M^n\subset\mathbb{R}^{n+2} , we characterize the vanishing of the normal curvature R_D at a point p \in M in terms of the behaviour of the asymptotic directions and the curvature locus at p . We relate the affine properties of codimension 2 submanifolds with flat normal bundle with the conformal properties of hypersurfaces in Euclidean space. We also characterize the semiumbilical, hypespherical and conformally flat submanifolds of codimension 2 in terms of their curvature loci.

Width and flow of hypersurfaces by curvature functions

In this paper, we generalize a well-known estimate of Colding-Minicozzi for the extinction time of convex hypersurfaces in Euclidean space evolving by their mean curvature to a much broader class of parabolic curvature flows.

Bernstein-Heinz-Chern results in calibrated manifolds

Given a calibrated Riemannian manifold \overline{M} with parallel calibration \Omega of rank m and M an orientable m-submanifold with parallel mean curvature H , we prove that if \cos\theta is bounded away from zero, where \theta is the \Omega -angle of M , and if M has zero Cheeger constant, then M is minimal. In the particular case M is complete with Ricci^M\geq 0 we may replace the boundedness condition on \cos\theta by \cos\theta\geq Cr^{-\beta} , when r\rightarrow+\infty , where 0 < \beta < 1 and C > 0 are constants and r is the distance function to a point in M . Our proof is surprisingly simple and extends to a very large class of submanifolds in calibrated manifolds, in a unified way, the problem started by Heinz and Chern of estimating the mean curvature of graphic hypersurfaces in Euclidean spaces. It is based on an estimation of \|H\| in terms of \cos\theta and an isoperimetric inequality. In a similar way, we also give some conditions to conclude M is totally geodesic. We study some particular cases.

Toeplitz operators on Bergman spaces with locally integrable symbols

Given a calibrated Riemannian manifold \overline{M} with parallel calibration \Omega of rank m and M an orientable m-submanifold with parallel mean curvature H , we prove that if \cos\theta is bounded away from zero, where \theta is the \Omega -angle of M , and if M has zero Cheeger constant, then M is minimal. In the particular case M is complete with Ricci^M\geq 0 we may replace the boundedness condition on \cos\theta by \cos\theta\geq Cr^{-\beta} , when r\rightarrow+\infty , where 0 < \beta < 1 and C > 0 are constants and r is the distance function to a point in M . Our proof is surprisingly simple and extends to a very large class of submanifolds in calibrated manifolds, in a unified way, the problem started by Heinz and Chern of estimating the mean curvature of graphic hypersurfaces in Euclidean spaces. It is based on an estimation of \|H\| in terms of \cos\theta and an isoperimetric inequality. In a similar way, we also give some conditions to conclude M is totally geodesic. We study some particular cases.

Rigidity theorems on hemispheres in non-positive space forms

We study the curvature condition which uniquely characterizes the hemisphere. In particular, we prove the Min-Oo conjecture for hypersurfaces in Euclidean space and hyperbolic space.

The volume-preserving mean curvature flow in Euclidean space

We study the convergence of the volume-preserving mean curvature flow of hypersurfaces in Euclidean space under some initial integral pinching conditions. We prove that if the traceless second fundamental form is sufficiently small, the flow will exist for all time and converge exponentially fast to a round sphere.

The holomorphic Gauss parametrization

We give a local parametric description of all complex hypersurfaces in $${\mathbb{C}^{n+1}}$$ and in complex projective space $${\mathbb{CP}^{n+1}}$$ with constant index of relative nullity, together with applications. This is a complex analogue to the parametrization for real hypersurfaces in Euclidean space known as the Gauss parametrization.

Gluing constructions amongst constant mean curvature hypersurfaces in $${\mathbb {S}^{n+1}}$$

Four constructions of constant mean curvature (CMC) hypersurfaces in \({\mathbb {S}^{n+1}}\) are given, which should be considered analogues of ‘classical’ constructions that are possible for CMC hypersurfaces in Euclidean space. First, Delaunay-like hypersurfaces, consisting roughly of a chain of hyperspheres winding multiple times around an equator, are shown to exist for all the values of the mean curvature. Second, a hypersurface is constructed which consists of two chains of spheres winding around a pair of orthogonal equators, showing that Delaunay-like hypersurfaces can be fused together in a symmetric manner. Third, a Delaunay-like handle can be attached to a generalized Clifford torus of the same mean curvature. Finally, two generalized Clifford tori of equal but opposite mean curvature of any magnitude can be attached to each other by symmetrically positioned Delaunay-like ‘arms’. This last result extends Butscher and Pacard’s doubling construction for generalized Clifford tori of small mean curvature.

Higher order Willmore hypersurfaces in Euclidean space

Let x: M n → R n+1 be an n(≥ 2)-dimensional hypersurface immersed in Euclidean space R n+1. Let σ i (0 ≤ i ≤ n) be the ith mean curvature and Q n = Σ i=0 n (−1)i+1( i n )σ 1 n−i σ i . Recently, the author showed that W n (x) = ∫ M Q n dM is a conformal invariant under conformal group of R n+1 and called it the nth Willmore functional of x. An extremal hypersurface of conformal invariant functional W n is called an nth order Willmore hypersurface. The purpose of this paper is to construct concrete examples of the 3rd order Willmore hypersurfaces in R 4 which have good geometric behaviors. The ordinary differential equation characterizing the revolutionary 3rd Willmore hypersurfaces is established and some interesting explicit examples are found in this paper.

Forced convex mean curvature flow in Euclidean spaces

We show the mean curvature flow of convex hypersurfaces in Euclidean spaces with a general forcing term may shrink to a point in finite time if the forcing term is small, or exist for all times and expand to infinity if the forcing term is large enough. The flow can converge to a round sphere in special cases. Long time existence and convergence of the normalization of the flow are studied.

Totally geodesic shadow boundary submanifolds and a characterization for $${\mathbb{S}}^n$$

We prove that the sphere \({\mathbb{S}}^n\) is the only compact immersed hypersurface in Euclidean space \({\mathbb{R}}^{n+1}\) each of whose shadow boundaries is a totally geodesic submanifold. Furthermore, we give conditions for the shadow boundary of a submanifold of \({\mathbb{R}}^{n+1}\) to be regular.

The lightlike flat geometry on spacelike submanifolds of codimension two in Minkowski space

We introduce the notion of the lightcone Gauss-Kronecker curvature for a spacelike submanifold of codimension two in Minkowski space, which is a generalization of the ordinary notion of Gauss curvature of hypersurfaces in Euclidean space. In the local sense, this curvature describes the contact of such submanifolds with lightlike hyperplanes. We study geometric properties of such curvatures and show a Gauss-Bonnet type theorem. As examples we have hypersurfaces in hyperbolic space, spacelike hypersurfaces in the lightcone and spacelike hypersurfaces in de Sitter space.

Embedding [formula omitted] into [formula omitted] with given integral Gauss curvature and optimal mass transport on [formula omitted

In [A.D. Aleksandrov, Convex Polyhedra, GITTL, Moscow, USSR, 1950 (in Russian); English translation: A.D. Aleksandrov, Convex Polyhedra, Springer, Berlin–New York, 2005] A.D. Aleksandrov raised a general question of finding variational statements and proofs of existence of polytopes with given geometric data. The first goal of this paper is to give a variational solution to the problem of existence and uniqueness of a closed convex hypersurface in Euclidean space with prescribed integral Gauss curvature. Our solution includes the case of a convex polytope. This problem was also first considered by Aleksandrov and below it is referred to as Aleksandrov's problem. The second goal of this paper is to show that in variational form the Aleksandrov problem is closely connected with the theory of optimal mass transport on a sphere with cost function and constraints arising naturally from geometric considerations.

Singularities of mean curvature flow and flow with surgeries

We collect in this paper several results on the formation of singularities in the mean curvature flow of hypersurfaces in euclidean space, under various kinds of convexity assumptions. We include some recent estimates for the flow of 2-convex surfaces, i.e. the surfaces where the sum of the two smallest principal curvatures is positive everywhere. Such results enable the construction of a flow with surgeries for these surfaces similar to the one introduced by Hamilton and Perelman for the Ricci flow. The topological applications of the construction are also described.

An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures

We study hypersurfaces in Euclidean space \(\mathbb{R}^{n+1}\) whose position vector x satisfies the condition L k x = Ax + b, where L k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed \(k=0,\ldots,n-1\), \(A \in \mathbb{R}^{(n+1)\times (n+1)}\) is a constant matrix and \(b\in\mathbb{R}^{n+1}\) is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature and open pieces of round hyperspheres and generalized right spherical cylinders of the form \(\mathbb{S}^{m}(r)\times\mathbb{R}^{n-m}\), with \(k+1\leq m \leq n-1\). This extends a previous classification for hypersurfaces in \(\mathbb{R}^{n+1}\) satisfying \(\Delta x=Ax+b\), where \(\Delta=L_{0}\) is the Laplacian operator of the hypersurface, given independently by Hasanis and Vlachos [J. Austral. Math. Soc. Ser. A 53, 377–384 (1991) and Chen and Petrovic [Bull. Austral. Math. Soc. 44, 117–129 (1991)].