Gauss-Kronecker Curvature Research Articles

A normal line congruence and minimal ruled Lagrangian submanifolds in [formula omitted

We characterize Lagrangian submanifolds in complex projective space for which each parallel submanifold along normal geodesics with respect to a unit normal vector field is Lagrangian, by using a normal line congruence of the Lagrangian submanifold to complex 2-plane Grassmannian and quaternionic Kähler structure. As a special case, we can construct minimal ruled Lagrangian submanifolds in complex projective space from an austere hypersurface in sphere with non-vanishing Gauss-Kronecker curvature.

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>

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.

A note on the Chern Conjecture in dimension four

Let M4 be a closed immersed minimal hypersurface with constant squared length of the second fundamental form S and constant 3-mean curvature H3 in S5. If H32≤12 and Gauss-Kronecker curvature KM satisfies KM≤1 (or KM≤S2144), then M4 is isoparametric.

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.

Parabolic hypersurfaces with constant mean curvature in Euclidean space

In this paper we consider $O(m)\times O(n)$-invariant parabolic hypersurfaces of Euclidean space with constant mean curvature. We analyse the orbit space of the $O(m)\times O(n)$-group on $\mathbb{R}^{m+n}$ to give some classification results of these hypersurfaces for which the Gauss-Kronecker curvature does not change its sign.

Nomizu–Smyth type results in the hyperbolic and de Sitter spaces via Gauss–Kronecker curvature

Nomizu–Smyth type results in the hyperbolic and de Sitter spaces via Gauss–Kronecker curvature

Gauss–Kronecker curvature and equisingularity at infinity of definable families

Assume given a polynomially bounded o-minimal structure expanding the real numbers. Let $(T_s)_{s\in \mathbb{R}}$ be a globally definable one parameter family of $C^2$-hypersurfaces of $\mathbb{R}^n$. Upon defining the notion of generalized critical value for such a family we show that the functions $s \to |K(s)|$ and $s\to K(s)$, respectively the total absolute Gauss-Kronecker and total Gauss-Kronecker curvature of $T_s$, are continuous in any neighbourhood of any value which is not generalized critical. In particular this provides a necessary criterion of equisingularity for the family of the levels of a real polynomial.

Translation hypersurfaces whose curvature depends partially on its variables

We investigate translation hypersurfaces in the (n+1)-dimensional Euclidean space whose Gauss-Kronecker curvature depends on either its first p or on its second q variables. These hypersurfaces are the graph of the sum of two functions in p and q independent variables respectively and with n=p+q. We prove that under this condition, the Gauss-Kronecker curvature is constant zero. On other side, if the mean curvature is nowhere zero and it depends on either its first p or on its second q variables, we get again that the Gauss-Kronecker curvature is constant zero.

On the Principal Curvatures of Complete Minimal Hypersurfaces in Space Forms

In recent decades, there has been an increase in the number of publications related to the hypersurfaces of real space forms with two principal curvatures. The works focus mainly on the case when one of the two principal curvatures is simple. The purpose of this paper is to study a slightly more general class of complete minimal hypersurfaces in real space forms of constant curvature c, namely those with $$\mathrm{n}-1$$ principal curvatures having the same sign everywhere. From assumptions on the scalar curvature R and the Gauss–Kronecker curvature K we characterize Clifford tori if $$c > 0$$ and prove that K is identically zero if $$c \le 0$$ .

A Characterization of Cylinders and an Estimate for Mean Curvature of Convex Euclidean Hypersurfaces Satisfying a Cylindrical Condition

We show that a curvature condition on the Gauss-Kronecker curvature and scalar curvature of a convex smooth hypersurface lying in the four dimensional Euclidean space yields a lower bound for the mean curvature. The curvature condition we investigate is suggested by the local geometry of cylinders in the four dimensional Euclidean space.

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.

Smooth solutions to the $$L_p$$ L p dual Minkowski problem

In this paper, we consider the $$L_p$$ dual Minkowski problem by geometric variational method. Using anisotropic Gauss–Kronecker curvature flows, we establish the existence of smooth solutions of the $$L_p$$ dual Minkowski problem when $$pq\ge 0$$ and the given data is even. If $$f\equiv 1$$ , we show under some restrictions on p and q that the only even, smooth, uniformly convex solution is the unit ball.

Convexity, Rigidity, and Reduction of Codimension of Isometric Immersions into Space Forms

We consider isometric immersions of complete connected Riemannian manifolds into space forms of nonzero constant curvature. We prove that if such an immersion is compact and has semi-definite second fundamental form, then it is an embedding with codimension one, its image bounds a convex set, and it is rigid. This result generalizes previous ones by do Carmo and Lima, as well as by do Carmo and Warner. It also settles affirmatively a conjecture by do Carmo and Warner. We establish a similar result for complete isometric immersions satisfying a stronger condition on the second fundamental form. We extend to the context of isometric immersions in space forms a classical theorem for Euclidean hypersurfaces due to Hadamard. In this same context, we prove an existence theorem for hypersurfaces with prescribed boundary and vanishing Gauss-Kronecker curvature. Finally, we show that isometric immersions into space forms which are regular outside the set of totally geodesic points admit a reduction of codimension to one.

Minimal hypersurfaces in with vanishing Gauss–Kronecker curvature

We give a local description of complete minimal hypersurfaces in with zero Gauss–Kronecker curvature, zero -mean curvature and nowhere-zero second fundamental form.

Remarks on hypersurfaces with constant higher order mean curvature in Euclidean space

In this paper we consider complete oriented hypersurfaces of Euclidean space with constant higher order mean curvature and having two principal curvatures, one of them simple. As an application of the so called principal curvature theorem, a purely geometric result on the principal curvatures of the hypersurface given by Smyth and Xavier (Invent Math 90:443–450, 1987), we characterize those hypersurfaces for which the Gauss–Kronecker curvature does not change sign, extending to the general n-dimensional case a previous result for surfaces due to Klotz and Osserman.

Spacelike submanifolds of codimension two in anti-de Sitter space

ABSTRACTIn this paper, we study spacelike submanifolds of codimension two in anti-de Sitter space from the viewpoint of Legendrian singularity theory. We introduce the notion of the anti-de Sitter normalized Gauss map which is a generalization of the ordinary notion of Gauss map of hypersurfaces in Euclidean space. We also introduce the AdS-normalized Gauss–Kronecker curvature for a spacelike submanifold of codimention two in anti-de Sitter space. In the local sense, this curvature describes the contact of submanifolds with some model surfaces.

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 $1$.