arXiv subjects
Publications and source records attributed to Toru Sasahara.
We give a complete classification of proper biharmonic Lagrangian $H$-umbilical submanifolds in complex space forms. Such submanifolds exist only in complex projective space and are locally congruent to an explicit model obtained via the Hopf fibration. In particular, their nonexistence in complex Euclidean space provides a partial affirmative answer to Chen's conjecture.
We study Hamiltonian stationary Lagrangian surfaces in complex space forms with nowhere-zero mean curvature vector $H$. We prove that if the Gaussian curvature is constant and $\grad|H|$ has positive constant length, then both the surface and the ambient space are flat. Moreover, the immersion is locally congruent to the known surface obtained as the product of a Cornu spiral and its reflection.
In this paper, we study Hamiltonian stationary Lagrangian surfaces in complex space forms. We first show that when the mean curvature is a non-zero constant, the second fundamental form is parallel. We then consider the case in which the mean curvature is a non-constant harmonic function. Under the additional assumption that the Gaussian curvature is constant, we obtain a complete classification of such Lagrangian surfaces.
We prove that a surface in Euclidean $3$-space has Maslovian normal bundle if and only if it is a part of a round sphere, a circular cylinder, or a circular cone.
The notion of Lagrangian $H$-umbilical submanifolds was introduced by B. Y. Chen in 1997, and these submanifolds have appeared in several important problems in the study of Lagrangian submanifolds from the Riemannian geometric point of view. Recently, the author introduced the notion of tangentially biharmonic submanifolds, which are defined as submanifolds such that the bitension field of the inclusion map has vanishing tangential component. The normal bundle of a round hypersphere in $\mathbb{R}^n$ can be immersed as a tangentially biharmonic Lagrangian $H$-umbilical submanifold in $\mathbb{C}^n$. Motivated by this fact, we classify tangentially biharmonic Lagrangian $H$-umbilical submanifolds in complex space forms.
In [5], D. Fetcu and C. Oniciuc presented the classification result for biharmonic $\mathcal{C}$-parallel Legendrian submanifolds in $7$-dimensional Sasakian space forms. However, it is incomplete. In this paper, all such submanifolds are explicitly determined.
In the late 1990s, B. Y. Chen introduced the notion of special slant surfaces in Kähler surfaces and classified non-minimal proper special slant surfaces with constant mean curvature in $2$-dimensional complex space forms. In this paper, we completely classify proper special slant surfaces with non-constant mean curvature in $2$-dimensional complex space forms.
It was proved in Chen's paper \cite{chen} that every real hypersurface in the complex projective plane of constant holomorphic sectional curvature $4$ satisfies $$ δ(2)\leq \frac{9}{4}H^2+5,$$ where $H$ is the mean curvature and $δ(2)$ is a $δ$-invariant introduced by him. In this paper, we study non-Hopf real hypersurfaces satisfying the equality case of the inequality under the condition that the mean curvature is constant along each integral curve of the Reeb vector field. We describe how to obtain all such hypersurfaces.
We investigate real hypersurfaces in complex space forms attaining equality in an inequality involving the contact $δ$-invariant $δ^c(2)$ introduced by Chen and Mihai in [3].
First, we classify proper biharmonic Hopf real hypersurfaces in $\mathbb{C}P^2$. Next, we classify proper biharmonic real hypersurfaces with two distinct principal curvatures in $\mathbb{C}P^n$, where $n\geq 2$. Finally, we prove that biharmonic ruled real hypersurfaces in $\mathbb{C}P^n$ are minimal, where $n\geq 2$.
We establish an inequality among the Ricci curvature, the squared mean curvature, and the normal curvature for real hypersurfaces in complex space forms. We classify real hypersurfaces in two-dimensional non-flat complex space forms which admit a unit vector field satisfying identically the equality case of the inequality.
We determine non-Hopf hypersurfaces with constant mean curvature in the complex projective plane which attain equality in a basic inequality between the maximum Ricci curvature and the squared mean curvature.
We characterize biharmonic anti-invariant surfaces in $3$-dimensional generalized $(κ, μ)$-manifolds with non-zero constant mean curvature by means of the scalar curvature of the ambient space and the mean curvature. In addition, we give a method for constructing infinity many examples of biharmonic submanifolds in a certain $3$-dimensional generalized $(κ, μ)$-manifold. Moreover, we determine $3$-dimensional generalized $(κ, μ)$-manifolds which admit a certain kind of proper biharmonic foliation.
This paper surveys some of the known results on $δ$-ideal CR submanifolds in complex space forms, the nearly Kähler $6$-sphere and odd dimensional unit spheres. In addition, the relationship between $δ$-ideal CR submanifolds and critical points of the $λ$-bienergy is mentioned. Some topics on variational problem for the $λ$-bienergy are also presented.
We obtain the explicit representation of Legendre surfaces in the unit $5$-sphere with harmonic mean curvature vector field, under the condition that the mean curvature function is constant along a certain special direction.
A submanifold is said to be tangentially biharmonic if the bitension field of the isometric immersion that defines the submanifold has vanishing tangential component. The purpose of this paper is to prove that a surface in Euclidean $3$-space has tangentially biharmonic normal bundle if and only if it is either minimal, a part of a round sphere, or a part of a circular cylinder.
In this paper, we introduce the notion of a quasi-biharmonic submanifold in a pseudo-Riemannian manifold and classify quasi-biharmonic marginally trapped Lagrangian surfaces in Lorentzian complex space forms.