Constant Gauss-Kronecker Curvature Research Articles

Hypersurfaces of constant Gauss–Kronecker curvature with Li-normalization in affine space

For convex hypersurfaces in the affine space \({\mathbb {A}}^{n+1}\) (\(n\ge 2\)), A.-M. Li introduced the notion of \(\alpha \)-normal field as a generalization of the affine normal field. By studying a Monge–Ampère equation with gradient blowup boundary condition, we show that regular domains in \({\mathbb {A}}^{n+1}\), defined with respect to a proper convex cone and satisfying some regularity assumption if \(n\ge 3\), are foliated by complete convex hypersurfaces with constant Gauss–Kronecker curvature relative to the Li-normalization. When \(n=2\), a key feature is that no regularity assumption is required, and the result extends our recent work about the \(\alpha =1\) case.

Rotational Hypersurfaces with Constant Gauss-Kronecker Curvature

The authors study rotational hypersurfaces with constant Gauss-Kronecker curvature in ℝn. They solve the ODE associated with the generating curve of such hypersurface using integral expressions and obtain several geometric properties of such hypersurfaces. In particular, they discover a class of non-compact rotational hypersurfaces with constant and negative Gauss-Kronecker curvature and finite volume, which can be seen as the higher-dimensional generalization of the pseudo-sphere.

Generalized translation hypersurfaces in Euclidean space

The graph of a function f defined in some open set of the Euclidean space of dimension ( p + q ) is said to be a generalized translation graph if f may be expressed as the sum of two independent functions ϕ and ψ defined in open sets of the Euclidean spaces of dimension p and q , respectively. In this paper we give a classification of the generalized translation graphs with constant mean curvature or constant Gauss–Kronecker curvature.

Boundary estimates of solutions for a class of Monge–Ampère equations

On a smooth strictly convex bounded domain in R2, we establish some boundary estimates of the solution to the complete hyperbolic affine sphere equation. On the unit ball in Rn, we also give some sharp estimates of the solutions of a class of Monge–Ampère equations. As an application, we prove the relative hyperbolic hypersurface constructed by solving two linked Monge–Ampère equations, has constant Gauss–Kronecker curvature and has bounded principal curvatures.

Hypersurfaces with Nonzero Constant Gauss–Kronecker Curvature in M n+1(±1)

We study hypersurfaces in a unit sphere and in a hyperbolic space with nonzero constant Gauss–Kronecker curvature and two distinct principal curvatures one of which is simple. Denoting by K the nonzero constant Gauss–Kronecker curvature of hypersurfaces, we obtain some characterizations of the Riemannian products $$ {S}^{n-1}(a)\times {S}^1\left(\sqrt{1-{a}^2}\right),\kern0.5em {a}^2=1/\left(1+{K}^{{\scriptscriptstyle \frac{2}{n-2}}}\right)\mathrm{or}\kern0.5em {S}^{n-1}(a)\times {H}^1\left(-\sqrt{1+{a}^2}\right),\kern0.5em {a}^2=1/\left({K}^{{\scriptscriptstyle \frac{2}{n-2}}}-1\right). $$

The equivariant Minkowski problem in Minkowski space

The classical Minkowski problem in Minkowski space asks, for a positive function $\phi$ on $\mathbb{H}^d$, for a convex set $K$ in Minkowski space with $C^2$ space-like boundary $S$, such that $\phi(\eta)^{-1}$ is the Gauss--Kronecker curvature at the point with normal $\eta$. Analogously to the Euclidean case, it is possible to formulate a weak version of this problem: given a Radon measure $\mu$ on $\mathbb{H}^d$ the generalized Minkowski problem in Minkowski space asks for a convex subset $K$ such that the area measure of $K$ is $\mu$. In the present paper we look at an equivariant version of the problem: given a uniform lattice $\Gamma$ of isometries of $\mathbb{H}^d$, given a $\Gamma$ invariant Radon measure $\mu$, given a isometry group $\Gamma_{\tau}$ of Minkowski space, with $\Gamma$ as linear part, there exists a unique convex set with area measure $\mu$, invariant under the action of $\Gamma_{\tau}$. The proof uses a functional which is the covolume associated to every invariant convex set. This result translates as a solution of the Minkowski problem in flat space times with compact hyperbolic Cauchy surface. The uniqueness part, as well as regularity results, follow from properties of the Monge--Amp\`ere equation. The existence part can be translated as an existence result for Monge--Amp\`ere equation. The regular version was proved by T.~Barbot, F.~B\'eguin and A.~Zeghib for $d=2$ and by V.~Oliker and U.~Simon for $\Gamma_{\tau}=\Gamma$. Our method is totally different. Moreover, we show that those cases are very specific: in general, there is no smooth $\Gamma_\tau$-invariant surface of constant Gauss-Kronecker curvature equal to $1$.

On affine translation surfaces in affine space

In this work we give a systemic study of affine translation surfaces in affine 3-dimensional space. Specifically, we obtain the complete classification of minimal affine translation surfaces. Moreover, we consider affine translation surfaces with some natural geometric conditions, such as constant affine mean curvature and constant Gauss–Kronecker curvature. Some characterization results with these geometric conditions are also obtained.

Monge-Ampère Systems with Lagrangian Pairs

The classes of Monge-Amp\`ere systems, decomposable and bi-decomposable Monge-Amp\`ere systems, including equations for improper affine spheres and hypersurfaces of constant Gauss-Kronecker curvature are introduced. They are studied by the clear geometric setting of Lagrangian contact structures, based on the existence of Lagrangian pairs in contact structures. We show that the Lagrangian pair is uniquely determined by such a bi-decomposable system up to the order, if the number of independent variables $\geq 3$. We remark that, in the case of three variables, each bi-decomposable system is generated by a non-degenerate three-form in the sense of Hitchin. It is shown that several classes of homogeneous Monge-Amp\`ere systems with Lagrangian pairs arise naturally in various geometries. Moreover we establish the upper bounds on the symmetry dimensions of decomposable and bi-decomposable Monge-Amp\`ere systems respectively in terms of the geometric structure and we show that these estimates are sharp (Proposition 4.2 and Theorem 5.3).

A characterization of quadric constant Gauss-Kronecker curvature hypersurfaces of spheres

A characterization of quadric constant Gauss-Kronecker curvature hypersurfaces of spheres

New characterizations for hyperbolic cylinders in anti-de Sitter spaces

In this paper we study complete maximal spacelike hypersurfaces in anti-de Sitter space H1n+1 with either constant scalar curvature or constant non-zero Gauss–Kronecker curvature. We characterize the hyperbolic cylinders Hm(c1)×Hn−m(c2),1≤m≤n−1, as the only such hypersurfaces with (n−1) principal curvatures with the same sign everywhere. In particular we prove that a complete maximal spacelike hypersurface in H15 with negative constant Gauss–Kronecker curvature is isometric to H1(c1)×H3(c2).

Affine Translation Surfaces with Constant Gaussian Curvature

Abstract. We study a–ne translation surfaces in R 3 and get a complete classiflcation ofsuch surfaces with constant Gauss-Kronecker curvature. 1. IntroductionA surface in E 3 is called a translation surface if it is obtained as a graph of a func-tion F ( x;y ) = p ( x )+ q ( y ), where p ( x ) and q ( y ) are difierentiable functions. It’s wellknown that a minimal translation surface in the Euclidean space E 3 must be a planeor a Scherk surface, which is the graph of the function F ( x;y ) = ln(cos x= cos y ),the only doubly periodic minimal translation surface.In this note, we study nondegenerate translation surfaces in a–ne space R 3 .This class of surfaces has been studied previously by many geometers. F. Manhart[3] classifled all the nondegenerate a–ne minimal translation surfaces in a–ne spaceR 3 . Further treatments are due to H. F. Sun [5], who classifled the nondegeneratea–ne translation surface with nonzero constant mean curvature in R 3 . Later on,Sun and Chen extended this into the case of hypersurfaces [6]. On the other hand,Binder [1] classifled locally symmetric a–ne translation surfaces in R

The Gauss–Kronecker curvature of minimal hypersurfaces in four-dimensional space forms

Let \({{\mathbb{Q}^4}(c)}\) be a four-dimensional space form of constant curvature c. In this paper we show that the infimum of the absolute value of the Gauss–Kronecker curvature of a complete minimal hypersurface in \({\mathbb{Q}^4(c), c\leq 0}\), whose Ricci curvature is bounded from below, is equal to zero. Further, we study the connected minimal hypersurfaces M 3 of a space form \({{\mathbb{Q}^4}(c)}\) with constant Gauss–Kronecker curvature K. For the case c ≤ 0, we prove, by a local argument, that if K is constant, then K must be equal to zero. We also present a classification of complete minimal hypersurfaces of \({\mathbb{Q}^4(c)}\) with K constant.

The affine complete hypersurfaces of constant Gauss-Kronecker curvature

Given a bounded convex domain Ω with C ∞ boundary and a function ϕ ∈ C ∞ (∂Ω), Li-Simon-Chen can construct an Euclidean complete and W -complete convex hypersurface M with constant affine Gauss-Kronecker curvature, and they guess the M is also affine complete. In this paper, we give a confirmation answer.

MAXIMAL SPACE-LIKE HYPERSURFACES IN H14(-1) WITH ZERO GAUSS-KRONECKER CURVATURE

In this paper, we study complete maximal space-like hypersurfaces with constant Gauss-Kronecker curvature in an antide Sitter space <TEX>$H_1^4(-1)$</TEX>. It is proved that complete maximal spacelike hypersurfaces with constant Gauss-Kronecker curvature in an anti-de Sitter space <TEX>$H_1^4(-1)$</TEX> are isometric to the hyperbolic cylinder <TEX>$H^2(c1){\times}H^1(c2)$</TEX> with S = 3 or they satisfy <TEX>$S{\leq}2$</TEX>, where S denotes the squared norm of the second fundamental form.

Curvatures of complete hypersurfaces in space forms

In this paper we investigate three-dimensional complete minimal hypersurfaces with constant Gauss-Kronecker curvature in a space form M4(c) (c ≤ 0). We prove that if the scalar curvature of a such hypersurface is bounded from below, then its Gauss-Kronecker curvature vanishes identically. Examples of complete minimal hypersurfaces which are not totally geodesic in the Euclidean space E4 and the hyperbolic space H4(c) with vanishing Gauss-Kronecker curvature are also presented. It is also proved that totally umbilical hypersurfaces are the only complete hypersurfaces with non-zero constant mean curvature and with zero quasi-Gauss-Kronecker curvature in a space form M4(c) (c ≤ 0) if the scalar curvature is bounded from below. In particular, we classify complete hypersurfaces with constant mean curvature and with constant quasi-Gauss-Kronecker curvature in a space form M4(c) (c ≤ 0) if the scalar curvature r satisfies r≥ ⅔c.

Rigidity theorems of Clifford Torus

Let M be an n-dimensional closed minimally immersed hypersurface in the unit sphere Sn + 1. Assume in addition that M has constant scalar curvature or constant Gauss-Kronecker curvature. In this note we announce that if M has (n - 1) principal curvatures with the same sign everywhere, then M is isometric to a Clifford Torus &lt;img src="http:/img/fbpe/aabc/v73n3/03ab.gif" alt="03ab.gif (725 bytes)" align="middle"&gt;.

Un lemme de Morse pour les surfaces convexes

We describe the "hyperbolic" properties of a riemann surface lamination M canonically associated to every compact three manifolds of curvature less than 1. More precisely, if the geodesic flow is the phase space attached to an ordinary differential equation, our space M is the "phase space" attached to a certain elliptic equation, which geometrically describes surfaces of constant Gauss-Kronecker curvature k (k-surfaces), 0<k<1. These hyperbolic properties are : a generic leaf is dense, the union of compact leaves is dense, and stability. They follow from an analogous of the Morse lemma for geodesics, which we call Morse lemma for convex surfaces, and which play the role of a shadowing lemma. We also prove theorems on existence and unicity of k-surfaces.

Conformally flat hypersurfaces with constant Gauss-Kronecker curvature

We consider 3-dimensional conformally flat hypersurfaces of E4 with constant Gauss-Kronecker curvature. We prove that those with three different principal curvatures must necessarily have zero Gauss-Kronecker curvature.

The Dirichlet problem for Monge-Ampère equations in non-convex domains and spacelike hypersurfaces of constant Gauss curvature

In this paper we extend the well known results on the existence and regularity of solutions of the Dirichlet problem for Monge-Ampère equations in a strictly convex domain to an arbitrary smooth bounded domain in $\mathbf {R}^n$ as well as in a general Riemannian manifold. We prove for the nondegenerate case that a sufficient (and necessary) condition for the classical solvability is the existence of a subsolution. For the totally degenerate case we show that the solution is in $C^{1,1} (\overline {\Omega })$ if the given boundary data extends to a locally strictly convex $C^2$ function on $\overline {\Omega }$. As an application we prove some existence results for spacelike hypersurfaces of constant Gauss-Kronecker curvature in Minkowski space spanning a prescribed boundary.

Spacelike hypersurfaces with constant Gauss-Kronecker curvature in the Minkowski space

Spacelike hypersurfaces with constant Gauss-Kronecker curvature in the Minkowski space