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Bang-Yen Chen

Publications and source records attributed to Bang-Yen Chen.

At least 55 records · Page 3Linked to original sources
arXiv2014

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].

math.DG↗
arXiv2014

Classification of ideal submanifolds of real space forms with type number $\leq 2$

Roughly speaking, an ideal immersion of a Riemannian manifold into a real space form is an isometric immersion which produces the least possible amount of tension from the ambient space at each point of the submanifold. The main purpose of this paper is to completely classify all non-minimal ideal submanifolds of real space forms with type number $\leq 2$.

math.DG↗
arXiv2013

Lagrangian submanifolds with prescribed second fundamental form

We classify Lagrangian submanifolds of complex space forms, whose second fundamental form can be written in a certain way, depending on a real parameter. For some special values of this parameter, the resulting submanifolds are ideal in the sense that they realize equality in an inequality for a Chen's delta-curvature.

math.DG↗
arXiv2013

Geometric classifications of homogeneous production functions

In this paper, we completely classify homogeneous production functions with an arbitrary number of inputs whose production hypersurfaces are flat. As an immediate consequence, we obtain a complete classification of homogeneous production functions with two inputs whose production surfaces are developable.

math.DG↗
arXiv2013

Classification of spherical Lagrangian submanifolds in complex Euclidean spaces

An isometric immersion $f:M^n\to \tilde M^n$ from a Riemannian $n$-manifold $M^n$ into a Kähler $n$-manifold $\tilde M^n$ is called {\it Lagrangian} if the complex structure $J$ of the ambient manifold $\tilde M^n$ interchanges each tangent space of $M^n$ with the corresponding normal space. In this paper, we completely classify spherical Lagrangian submanifolds in complex Euclidean spaces. Furthermore, we also provide two corresponding classification theorems for Lagrangian submanifolds in the complex pseudo-Euclidean spaces with arbitrary complex index.

math.DG↗
arXiv2013

Totally geodesic submanifolds of symmetric spaces, III

One purpose of this article is to establish a general method to determine stability of totally geodesic submanifolds of symmetric spaces. The method is used to determine the stability of the basic totally geodesic submanifolds $M_+,M_-$ introduced and studied by Chen and Nagano in [Totally geodesic submanifolds of symmetric spaces, II, Duke Math. J. 45 (1978), 405--425] as minimal submanifolds. The other purpose is to establish a stability theorem for minimal totally real submanifolds of Kählerian manifolds.

math.DG↗
arXiv2013

Minimal flat Lorentzian surfaces in Lorentzian complex space forms

In this article we study minimal flat Lorentzian surfaces in Lorentzian complex space forms. First we prove that, for minimal flat Lorentzian surfaces in a Lorentzian complex form, the equation of Ricci is a consequence of the equations of Gauss and Codazzi. Then we classify minimal flat Lorentzian surfaces in the Lorentzian complex plane ${\bf C}^2_1$. Finally, we classify minimal flat slant surfaces in Lorentzian complex projective plane $CP^2_1$ and in Lorentzian complex hyperbolic plane $CH^2_1$.

math.DG↗
arXiv2013

Open problems and conjectures on submanifolds of finite type revisited

Submanifolds of finite type were introduced by the author during the late 1970s. The first results on this subject had been collected in author's book [Total mean curvature and sub manifolds of finite type, World Scientific, NJ, 1984]. A list of ten open problems and three conjectures on submanifolds of finite type was published in 1981. The main purpose of this article is to provide some updated information on the three conjectures.

math.DG↗
arXiv2013

Ideal hypersurfaces of Euclidean four-space

The notion of ideal immersions was introduced by the author in 1990s. Roughly speaking, an ideal immersion of a Riemannian manifold into a real space form is a nice isometric immersion which produces the least possible amount of tension from the ambient space at each point. In this paper, we classify all ideal hypersurfaces with two distinct principal curvatures in the Euclidean 4-space $\mathbb E^4$. Moreover, we prove that such ideal hypersurfaces are always rigid. Furthermore, we show that non-minimal ideal hypersurfaces with three distinct principal curvatures in $\mathbb E^4$ are also rigid. On the other hand, we provide explicit examples to illustrate that minimal ideal hypersurfaces with three principal curvatures in $\mathbb E^4$ are not necessarily rigid.

math.DG↗
arXiv2013

A survey on Geometry of Slant Submanifolds

The present volume is the written version of the series of lectures the author delivered at the Catholic University of Leuven, Belgium during the period of June-July, 1990. The main purpose of these talks is to present some of author's work and also his joint works with Professor T. Nagano and Professor Y. Tazawa of Japan, Professor P. F. Leung of Singapore and Professor J. M. Morvan of France on geometry of slant submanifolds and its related subjects in a systematical way.

math.DG↗
arXiv2013

Lagrangian submanifolds in complex space forms satisfying an improved equality involving $δ(2,2)$

It was proved in [8,9] that every Lagrangian submanifold $M$ of a complex space form $\tilde M^{5}(4c)$ of constant holomorphic sectional curvature $4c$ satisfies the following optimal inequality: {align}\tag{A}δ(2,2)\leq \text{\small${25}{4}$} H^{2}+8c,{align} where $H^{2}$ is the squared mean curvature and $δ(2,2)$ is a $δ$-invariant on $M$ introduced by the first author. This optimal inequality improves a special case of an earlier inequality obtained in [B.-Y. Chen, Japan. J. Math. 26 (2000), 105-127]. The main purpose of this paper is to classify Lagrangian submanifolds of $\tilde M^{5}(4c)$ satisfying the equality case of the improved inequality (A).

math.DG↗
arXiv2013

Classification of minimal Lorentz surfaces in indefinite space forms with arbitrary codimension and arbitrary index

Since J. L. Lagrange initiated in 1760 the study of minimal surfaces of Euclidean 3-space, minimal surfaces in real space forms have been studied extensively by many mathematicians during the last two and half centuries. In contrast, so far very few results on minimal Lorentz surfaces in indefinite space forms are known. Hence, in this paper we investigate minimal Lorentz surfaces in arbitrary indefinite space forms. As a consequence, we obtain several classification results for minimal Lorentz surfaces in indefinite space forms. In particular, we completely classify all minimal Lorentz surfaces in a pseudo-Euclidean space $\mathbb E^m_s$ with arbitrary dimension $m$ and arbitrary index $s$.

math.DG↗
arXiv2013

Dependence of the Gauss-Codazzi equations and the Ricci equation of Lorentz surfaces

The fundamental equations of Gauss, Codazzi and Ricci provide the conditions for local isometric embeddability. In general, the three fundamental equations are independent for surfaces in Riemannian 4-manifolds. In contrast, we prove in this article that for arbitrary Lorentz surfaces in Lorentzian Kaehler surfaces the equation of Ricci is a consequence of the equations of Gauss and Codazzi.

math.DG↗
arXiv2013

Geometry of quasi-sum production functions with constant elasticity of substitution property

A production function $f$ is called quasi-sum if there are strict monotone functions $F, h_1,...,h_n$ with $F'>0$ such that $$f(x)= F(h_1 (x_1)+...+h_n (x_n)).$$ The justification for studying quasi-sum production functions is that these functions appear as solutions of the general bisymmetry equation and they are related to the problem of consistent aggregation. In this article, first we present the classification of quasi-sum production functions satisfying the constant elasticity of substitution property. Then we prove that if a quasi-sum production function satisfies the constant elasticity of substitution property, then its graph has vanishing Gauss-Kronecker curvature (or its graph is a flat space) if and only if the production function is either a linearly homogeneous generalized ACMS function or a linearly homogeneous generalized Cobb-Douglas function.

math.OC↗
arXiv2013

Construction of Hamiltonian-stationary Lagrangian submanifolds of constant curvature $\varepsilon$ in complex space forms $\tilde M^n(4\varepsilon)

Lagrangian submanifolds of a Kaehler manifold are called Hamiltonian-stationary (or $H$-stationary for short) if it is a critical point of the area functional restricted to compactly supported Hamiltonian variations. In [B. Y. Chen, F. Dillen, L. Verstraelen and L. Vrancken, Lagrangian isometric immersions of a real-space-form $M^{n}(c)$ into a complex-space-form $\tilde{M}^{n}(4c)$, Math. Proc. Cambridge Philo. Soc. 124 (1998), 107-125], an effective method to constructing Lagrangian submanifolds of constant curvature $\varepsilon$ in complex space form $M^n(4\varepsilon)$ was introduced. In this article we survey recent results on construction of Hamiltonian-stationary Lagrangian submanifolds in complex space forms using this method.

math.AP↗
arXiv2013

Recent developments of biharmonic conjecture and modified biharmonic conjectures

A submanifold $M$ of a Euclidean $m$-space is said to be biharmonic if $Δ\overrightarrow H=0$ holds identically, where $\overrightarrow H$ is the mean curvature vector field and $Δ$ is the Laplacian on $M$. In 1991, the author conjectured that every biharmonic submanifold of a Euclidean space is minimal. The study of biharmonic submanifolds is nowadays a very active research subject. In particular, since 2000 biharmonic submanifolds have been receiving a growing attention and have become a popular subject of study with many progresses. In this article, we provide a brief survey on recent developments concerning my original conjecture and generalized biharmonic conjectures. At the end of this article, I present two modified conjectures related with biharmonic submanifolds.

math.DG↗
arXiv2013

A Wintgen type inequality for surfaces in 4D neutral pseudo-Riemannian space forms and its applications to minimal immersions

Let $M$ be a space-like surface immersed in a 4-dimensional pseudo-Riemannian space form $R^4_2(c)$ with constant sectional curvature $c$ and index two. In the first part of this article, we prove that the Gauss curvature $K$, the normal curvature $K^D$, and mean curvature vector $H$ of $M$ satisfy the general inequality: $K+K^D\geq H,H +c$. In the second part, we investigate space-like minimal surfaces in $R^4_2(c)$ which satisfy the equality case of the inequality identically. Several classification results in this respect are then obtained.

math.DG↗