Submanifolds Of Finite Type Research Articles

Differential Geometry of 1-type Submanifolds and Submanifolds with 1-type Gauss Map

The theory of finite type submanifolds was introduced by the first author in late 1970s and it has become a useful tool for investigation of submanifolds. Later, the first author and P. Piccinni extended the notion of finite type submanifolds to finite type maps of submanifolds; in particular, to submanifolds with finite type Gauss map. Since then, there have been rapid developments in the theory of finite type. The simplest finite type submanifolds and submanifolds with finite type Gauss maps are those which are of 1-type. The classes of such submanifolds constitute very special and interesting families in the finite type theory.

Linear Weingarten $$\lambda $$-biharmonic hypersurfaces in Euclidean space

In order to derive the best possible estimates of the total mean curvature of a compact submanifold in a Euclidean space in terms of spectral geometry, in the late 1970s, Bang-Yen Chen introduced the theory of finite-type submanifolds, which could be viewed as $$\lambda $$ -biharmonic submanifolds in the sense of $$\lambda $$ -biharmonic maps. Interestingly, Chen proposed in 1991 the following problem (Chen in Soochow J Math 17:2:169–188, 1991, Problem 12): “Determine all submanifolds of Euclidean spaces which are of null 2-type. In particular, classify null 2-type hypersurfaces in Euclidean spaces.” In this paper, we give further support evidence to the above problem. We are able to prove that a linear Weingarten null 2-type or $$\lambda $$ -biharmonic hypersurface $$M^n$$ of a Euclidean space $$\mathbb R^{n+1}$$ has constant mean curvature and constant scalar curvature provided $$n<7$$ .

A Note on Marcinkiewicz Integrals along Submanifolds of Finite Type

We study the parametric Marcinkiewicz integrals along submanifolds of finite type with rough kernels. The kernels of our operators are allowed to be very rough both on the unit sphere and in the radial direction. Under the rather weakened size conditions on the integral kernels, the Lp bounds will be established for such operators. As applications, the corresponding results for parametric Marcinkiewicz integrals related to area integrals and Littlewood-Paley gλ⁎ functions are also given.

Formula omitted]-ideal null 2-type hypersurfaces in Euclidean spaces

In the theory of finite type submanifolds, null 2-type submanifolds are the most simple ones, besides 1-type submanifolds (cf. e.g., [3,12]). In particular, the classification problems of null 2-type hypersurfaces are quite interesting and fundamentally important. In this paper, we prove that every δ(3)-ideal null 2-type hypersurface in a Euclidean space has constant mean curvature and constant scalar curvature.

Some open problems and conjectures on submanifolds of finite type: recent development

Submanifolds of finite type were introduced by the author during the late 1970s. The first results on this subject were collected in author's books [26,29]. In 1991, a list of twelve open problems and three conjectures on finite type submanifolds was published in [40]. A detailed survey of the results, up to 1996, on this subject was given by the author in [48]. Recently, the study of finite type submanifolds, in particular, of biharmonic submanifolds, have received a growing attention with many progresses since the beginning of this century. In this article, we provide a detailed account of recent development on the problems and conjectures listed in [40].

CLASSIFICATIONS OF HELICOIDAL SURFACES WITH L1-POINTWISE 1-TYPE GAUSS MAP

Abstract. In this paper, we study rotational and helicoidal surfaces inEuclidean 3-space in terms of their Gauss map. We obtain a completeclassification of these type of surfaces whose Gauss maps Gsatisfy L 1 G=f(G+C) for some constant vector C ∈E 3 and smooth function f, whereL 1 denotes the Cheng-Yau operator. 1. IntroductionLet Mbe a hypersurface of the (n+1)-dimensional Euclidean space E n+1 .A smooth mapping φ: M→ E N is said to be of k-type if it can be expressedas a sum of eigenvectors of Laplace operator ∆ corresponding to kdistincteigenvalues of ∆ ([6]). If φis an immersion from M into E n+1 is of k-type,then the submanifold Mis said to be of k-type ([3]). A good survey on finitetype submanifolds is [4].On the other hand, if the Gauss map Gof Msatisfies(1.1) ∆G= λ(G+C)for a constant λ∈ Rand a constant vector C, Mis said to have 1-type Gaussmap, [7]. However, the Gauss map of some important submanifolds such as ahelicoid and a catenoid in E 3 satisfies a very similar equation to (1.1), namely,(1.2) ∆G= f(G+C)for a smooth function f ∈ C

SOME CLASSIFICATION RESULTS ON FINITE-TYPE RULED SUBMANIFOLDS IN A LORENTZ-MINKOWSKI SPACE

Ruled submanifolds of finite type in Lorentz-Minkowski space are studied. We construct a new example of ruled submanifolds with degenerate rulings called a $BS$-kind ruled submanifold, which is of finite type. Also, it is determined by the restricted minimal polynomial of the shape operator associated with the mean curvature vector field.

Formula omitted] boundedness of Marcinkiewicz integral operators

In this paper we prove the L p boundedness of Marcinkiewicz integral operators associated to compact submanifolds of finite type under the L ( log L ) 1 / 2 condition on the kernel functions. The exponent 1/2 is optimal. We also show that the L p boundedness may fail to hold if the underlying submanifold is not required to be of finite type.

Rough Singular Integrals with Kernels Supported by Submanifolds of Finite Type

Rough Singular Integrals with Kernels Supported by Submanifolds of Finite Type

The tangential variation on hyperruled surfaces

One of the most interesting and profound aspects of classical differential geometry is its interplay with the calculus of variations. In fact, the main differential geometric ideas of the calculus of variation occur over and over again and are continually being invented and rediscovered in a vast array of classical and modern differential geometry. So the normal variational problem on general surfaces and hyperruled surfaces were studied by some geometers, specifically one may cite, in this regard, the works of Abdel-All and Abd-Ellah [Chaos, Solitons & Fractals, in press; Ind. J. Pure Appl. Math., in press; Adv. Model. Anal. 37 (2000)], Abdel-All and Hussien [J. Korean Math. Soc. 3 (1999) 663], Chen [J. Lond. Math. Soc. 6 (1973) 321; Total Mean Curvature and Submanifolds of Finite Type, World Scientific Publishing copre Ltd, 1984; Geometry of Submanifolds, Marcel Dekker, New York, 1973], Soliman et al. [Bull. Fac. Sci., Assiut Univ. 24 (1995) 189], Willmore [Total Curvature in Riemannian Geometry, Ellis Horwod Limited Publishers, 1982]. The purpose of the present work is to study effectiveness the tangential variation in the direction of the tangent for the base curve of hyperruled surfaces in E n+1 .

Singular integrals along submanifolds of finite type.

Theorem A. Let T8 be given as above. Suppose that (i) 8 is of finite type at 0, and (ii) ∈C1(Sn−1). Then, for 1 0 such that ‖T8f ‖Lp(Rd ) ≤ Cp‖f ‖Lp(Rd ) (1.4) for every f ∈Lp(Rd). It is well known that T8 may fail to be bounded on L for any p if condition (i) is removed (the precise definition of a finite type mapping will be reviewed in the next section). The purpose of this paper is to establish the L boundedness of T8 when condition (ii) is replaced by the following weaker condition: (ii′ ) ∈Lq(Sn−1) for some q > 1. This yields the following theorem.

On the fourier transform of measures carried by submanifolds of finite type

On the fourier transform of measures carried by submanifolds of finite type

On the Gauss map of ruled surfaces in a 3-dimensional minkowski space

Relative to Takahashi's theorem [9] for minimal submanifolds, the idea of submanifolds of finite type in a Euclidean space was introduced by Chen [2] and the theory is recently greatly developed. Let x : M^Rn+1 be an isometric immersion of n-dimensional Riemannian manifold into an (n+l)-dimensional Euclidean space Rn+1 and A the Laplacian on M. As a generalization of Takahashi's theorem for the case of hypersurfaces, Garay [4] considered the hypersurface satisfying the condition Ax=Ax, where A denotes the constant diagonal matrix of order n + 1. On the other hand, let x : M~^Rm be an isometric immersion of a compact oriented ^-dimensional Riemannian manifold into Rm. For a generalized Gauss

The Chen-type of the noncompact cyclides of Dupin

Minimal submanifolds of a Euclidean space are contained in a much larger class of submanifolds, namely in the class of submanifolds of finite type. Submanifolds of finite type were introduced about a decade ago by B. Y. Chen in [2]; the first results on this subject have been collected in the books [2], [3].

Ruled Submanifolds of Finite Type

Ruled Submanifolds of Finite Type

Ruled submanifolds of finite type

We show that a ruled submanifold of finite type in a Euclidean space is a cylinder on a curve of finite type or a generalized helicoid.

BIHARMONIC W-SURFACES IN 4-DIMENSIONAL PSEUDO-EUCLIDEAN SPACE

In this article, W-surface in a 4-dimensional pseudo-Euclidean space Et4 (t = l, 2) is defind and a classification problem of biharmonic W-surfaces in Et4 is investigated. The study of biharmonic submanifold in a pseudo-Euclidean space is an off-spring of the study of a problem “ Classify all of finite type submanifolds in a Euclidean space Em” proposed by B. Y. Chen (see [5],[6] and [11]). As a result, one classification theorem for biharmonic W-surfaces with flat normal connection in Et4 is obtained. It is a generalization of main theorem in [7].

Finite type pseudo-umbilical submanifolds in a hypersphere

The notion of finite type submanifolds was introduced by B.Y. Chen. In this article we study 2- and 3-type pseudo-umbilical submanifolds in a hypersphere. Two theorems in this respect are obtained.

Finite type ruled manifolds shaped on spherical submanifolds

Euclidean submanifolds of finite type were introduced few years ago by B.-Y.Chen, [2], [3]. Since then there has been an increasingly interesting development of this subject, where one can observe that most papers concern to compact finite type submanifolds. This is because in such case the finite type structure can be characterized by a practical condition on the Laplacian of the mean curvature vector (see S1), which turns out to be only a necessary condition when the submanifold is non-compact. Frequently, the submanifolds constructed over a given submanifold are non-compact. For example, given a compact submanifold M of the unit sphere Sn+1 centered at the origin in En+2, one constructs the punctured cone CM-{0} shaped on M. In [8], J.Simons proved that if M is minimal in Sn+1 then CM-{0} is minimal in En+2. In the finite type submanifolds terminology, see [9], Simons’result says that if M is of 1-type then CM-{0} is also of 1-type. In that context, the following problem arises in a natural way.

Coordinate finite-type submanifolds

A submanifold of a Euclidean space is called a coordinate finite-type submanifold if its coordinate functions are eigenfunctions of Δ. We prove that the compact coordinate finite-type submanifolds are minimal submanifolds of quadratic hypersurfaces of Euclidean spaces. Moreover, we classify the compact coordinate finite-type submanifolds of codimension 2.