A class of minimal Lagrangian submanifolds in complex hyperquadrics

In this paper we construct a class of compact minimal Lagrangian submanifolds in complex hyperquadrics by studying Gauss maps of compact rotational hypersurfaces in the unit sphere.     

Embedded Rotational Hypersurfaces with Constant Scalar Curvature in Sn: A Correction to a Statement in M. L. Leite, Manuscripta Math. 67 (1990), 285-304.

We prove that there exist (n − 1)-dimensional compact embedded rotational hypersurfaces with constant scalar curvature (n − 1)(n − 2)S (S > 1) of S ⁿ other than product of spheres for 4 ≤ n ≤ 6. As a result, we prove that Leite’s Assertion was incorrect.The project is supported by the grant No. 10531090 of NSFC.     

The scalar curvature of CR submanifolds of maximal CR dimension of complex projective space

We treat n-dimensional compact minimal submanifolds of complex projective space when the maximal holomorphic tangent subspace is (n − 1)-dimensional and we give a sufficient condition for such submanifolds to be tubes over totally geodesic complex subspaces. Authors’ addresses: Mirjana Djorić, Faculty of Mathematics, University of Belgrade, Studentski trg 16, pb. 550, 11000 Belgrade, Serbia; Masafumi Okumura, 5-25-25 Minami Ikuta, Tama-ku, Kawasaki, Japan     

On spectral geometry of minimal parallel submanifolds

We consider the problem of characterizing some minimal submanifolds using the spectrum⁰Spec of the Laplace-Beltrami operator acting on fucntions. In particular we characterize then-dimensional compact minimal totally real parallel submanifolds immersed in the complex projective spaceCP ⁿ, 3≤n≤6, by their⁰Spec in the class of all compact totally real minimal submanifolds ofCP ⁿ. Moreover, we characterize the Clifford torus by its⁰Spec in the class of all compact minimal submanifolds of the Euclidean sphereS ⁿ⁺¹(1). Authors supported by funds of the University of Lecce and the M.U.R.S.T.     

A finiteness theorem for isoparametric hypersurfaces

In this note we prove that for eachn there are only finitely many diffeomorphism classes of compact isoparametric hypersurfaces ofS ⁿ⁺¹ with four distinct principal curvatures.     

Improved Chen–Ricci inequality for curvature-like tensors and its applications

We present Chen–Ricci inequality and improved Chen–Ricci inequality for curvature like tensors. Applying our improved Chen–Ricci inequality we study Lagrangian and Kaehlerian slant submanifolds of complex space forms, and C-totally real submanifolds of Sasakian space forms.     

Lower order eigenvalues of Dirichlet Laplacian

In this paper, we investigate an eigenvalue problem for the Dirichlet Laplacian on a domain in an n-dimensional compact Riemannian manifold. First we give a general inequality for eigenvalues. As one of its applications, we study eigenvalues of the Laplacian on a domain in an n-dimensional complex projective space, on a compact complex submanifold in complex projective space and on the unit sphere. By making use of the orthogonalization of Gram–Schmidt (QR-factorization theorem), we construct trial functions. By means of these trial functions, estimates for lower order eigenvalues are obtained. Qing-Ming Cheng research was partially supported by a Grant-in-Aid for Scientific Research from JSPS. Hejun Sun and Hongcang Yang research were partially supported by NSF of China.     

Pseudo-parallel Lagrangian submanifolds in complex space forms

In this work we study pseudo-parallel Lagrangian submanifolds in a complex space form. We give several general properties of pseudo-parallel submanifolds. For the 2-dimensional case, we show that any minimal Lagrangian surface is pseudo-parallel. We also give examples of non-minimal pseudo-parallel Lagrangian surfaces. Here we prove a local classification of the pseudo-parallel Lagrangian surfaces. In particular, semi-parallel Lagrangian surfaces are totally geodesic or flat. Finally, we give examples of pseudo-parallel Lagrangian surfaces which are not semi-parallel.     

Ruled Real Hypersurfaces in a Nonflat Quaternionic Space Form

In this paper we construct many ruled real hypersurfaces in a nonflat quaternionic space form systematically, and in particular give an example of a homogeneous ruled real hypersurface in a quaternionic hyperbolic space. In the second half of this paper we characterize them by investigating the extrinsic shape of their geodesics. We also characterize curvature-adapted real hypersurfaces in nonflat quaternionic space forms from the same viewpoint.The first author was partially supported by Grant-in-Aid for Scientific Research (C) (No. 14540075), Ministry of Education, Science, Sports and Culture.The second author was partially supported by Grant-in-Aid for Scientific Research (C) (No. 14540080), Ministry of Education, Science, Sports and Culture.     

Submanifolds of Weyl Flat Manifolds

 We consider compact Weyl submanifolds of Weyl flat manifolds with special attention on compact Einstein-Weyl hypersurfaces. In particular, in the last part of the paper, we study Weyl submanifolds of special noncompact manifolds, called PC-manifolds. Received July 16, 2001; in revised form February 6, 2002 Published online August 9, 2002     

Willmore Lagrangian Spheres in the Complex Euclidean Space C n

In this paper we construct many examples of n-dimensionalWillmore Lagrangian submanifolds in the complex Euclidean space C ⁿ . We characterize them as the only Willmore Lagrangian submanifolds invariant under the action of SO(n). The mostimportant contribution of our construction is that it provides examplesof Willmore Lagrangian spheres in C ⁿ for all n 2.     

Spherical submanifolds in a Euclidean space

In this paper, we are interested in extending the study of spherical curves in R ³ to the submanifolds in the Euclidean space R ^n+p . More precisely, we are interested in obtaining conditions under which an n-dimensional compact submanifold M of a Euclidean space R ^n+p lies on the hypersphere S ^n+p−1(c) (standardly imbedded sphere in R ^n+p of constant curvature c). As a by-product we also get an estimate on the first nonzero eigenvalue of the Laplacian operator Δ of the submanifold (cf. Theorem 3.5) as well as a characterization for an n-dimensional sphere S ⁿ (c) (cf. Theorem 4.1).     

The relative volume growth of minimal submanifolds

The volume growth of certain well-defined subsets of minimal submanifolds in riemannian spaces are compared with the volume growth of balls and spheres in space forms of constant curvature. Work partially supported by a DGICYT Grant No. PB97-1425 and Fundacio Caixa Castello Grant No. 0I027.     

A sharp height estimate for compact spacelike hypersurfaces with constant r-mean curvature in the Lorentz-Minkowski space and application

In this paper we obtain a sharp height estimate concerning compact spacelike hypersurfaces Σn immersed in the (n+1)-dimensional Lorentz-Minkowski space Ln+1 with some nonzero constant r-mean curvature, and whose boundary is contained into a spacelike hyperplane of Ln+1. Furthermore, we apply our estimate to describe the nature of the end of a complete spacelike hypersurface of Ln+1.     

Scalar curvature of a certain CR-submanifold of complex projective space

We study an n-dimensional, compact, minimal CR-submanifold of CR-dimension n − 1 and give a sufficient condition for the submanifold to be a tube over a totally geodesic complex subspace. Dedicated to Professor U-Hang Ki on his 60th birthday This work is supported by the research grant of the Catholic University of Taegu-Hyosung in 1996.     

Conformally flat,minimal, Lagrangian submanifolds in complex space forms

We investigate n-dimensional (n ⩾ 4), conformally flat, minimal, Lagrangian submanifolds of the n-dimensional complex space form in terms of the multiplicities of the eigenvalues of the Schouten tensor and classify those that admit at most one eigenvalue of multiplicity one. In the case where the ambient space is ℂⁿ, the quasi umbilical case was studied in Blair (2007). However, the classification there is not complete and several examples are missing. Here, we complete (and extend) the classification and we also deal with the case where the ambient complex space form has non-vanishing holomorphic sectional curvature.

     

Isoparametric hypersurfaces in Finsler space forms

In this paper, we study isoparametric hypersurfaces in Finsler space forms by investigating focal points, tubes and parallel hypersurfaces of submanifolds. We prove that the focal submanifolds of isoparametric hypersurfaces are anisotropic-minimal and obtain a general Cartan-type formula in a Finsler space form with vanishing reversible torsion, from which we give some classifications on the number of distinct principal curvatures or their multiplicities.