Parallel Mean Curvature Research Articles

Rigidity of Minimal Submanifolds in Space Forms

In this paper, we consider the rigidity for an $n(\geq 4)$-dimensional submanfolds $M^n$ with parallel mean curvature in the space form ${\mathbb M}^{n+p}_c$ when the integral Ricci curvature of $M$ has some bound. We prove that, if $c+H^2>0$ and $\|\mathrm{Ric}_{-}^\lambda\|_{n/2}< \epsilon(n,c, \lambda, H)$ for $\lambda$ satisfying $ \frac{n-2}{n-1} (c+H^2) < \lambda \le c+H^2$, then $M$ is the totally umbilical sphere $\mathbb{S}^n(\tfrac{1}{\sqrt{c+H^2}})$. Here $H$ is the norm of the parallel mean curvature of $M$, and $\epsilon(n,c,\lambda, H)$ is a positive constant depending only on $n, c,\lambda$ and $H$. This extends some of the earlier work of [15] from pointwise Ricci curvature lower bound to inetgral Ricci curvature lower bound.

Pointwise Slant Curves in Quasi-paraSasakian 3-Manifolds

This paper is devoted to the study of pointwise slant Frenet curves in a quasi-paraSasakian metric 3-manifold. We give a characterization result for the existence of such curves and determine their curvature and torsion in this class of manifold. Further, the characterizations for these curves having harmonic and $$\mathbf {C}$$ -parallel mean curvature vector field are derived.

Geometric Study of Marginally Trapped Surfaces in Space Forms and Robertson-Walker Spacetimes—An Overview

A marginally trapped surface in a spacetime is a Riemannian surface whose mean curvature vector is lightlike at every point. In this paper we give an up-to-date overview of the differential geometric study of these surfaces in Minkowski, de Sitter, anti-de Sitter and Robertson-Walker spacetimes. We give the general local descriptions proven by Anciaux and his coworkers as well as the known classifications of marginally trapped surfaces satisfying one of the following additional geometric conditions: having positive relative nullity, having parallel mean curvature vector field, having finite type Gauss map, being invariant under a one-parameter group of ambient isometries, being isotropic, being pseudo-umbilical. Finally, we provide examples of constant Gaussian curvature marginally trapped surfaces and state some open questions.

Rigidity of submanifolds with parallel mean curvature in a hyperbolic space

Rigidity of submanifolds with parallel mean curvature in a hyperbolic space

Self-shrinker Type Submanifolds in the Euclidean Space

In this paper, we devote to investigating a new class of submanifolds, named the self-shrinker type submanifolds. We define a functional and prove its critical point is a self-shrinker type submanifold. We also show that a compact self-shrinker type submanifold with the parallel mean curvature vector in the Euclidean Space $$\mathbb {R}^{n+p}$$ is a minimal submanifold in a hypersphere $$\mathbb {S}^{n+p-1}$$ of $$\mathbb {R}^{n+p}$$ .

Generalized Ejiri’s Rigidity Theorem for Submanifolds in Pinched Manifolds

Let Mn(n ≥ 4) be an oriented compact submanifold with parallel mean curvature in an (n + p)-dimensional complete simply connected Riemannian manifold Nn+p. Then there exists a constant δ(n, p) 2 (0, 1) such that if the sectional curvature of N satisfies $${\overline K _N} \in \;\,\left[ {\delta \left( {n,p} \right),\;1} \right]$$, and if M has a lower bound for Ricci curvature and an upper bound for scalar curvature, then N is isometric to Sn+p. Moreover, M is either a totally umbilic sphere $${S^n}({1 \over {\sqrt {1 + {H^2}} }})$$, a Clifford hypersurface $${S^m}({1 \over {\sqrt {2(1 + {H^2})} }})\; \times \;{S^m}({1 \over {\sqrt {2(1 + {H^2})} }})$$ in the totally umbilic sphere $${S^{n+1}}({1 \over {\sqrt {1 + {H^2}} }})$$ with n = 2m, or $${\mathbb{C}{\rm{P}}^2}\left( {{4 \over 3}\left( {1 + {H^2}} \right)} \right)$$ in $${S^7}({1 \over {\sqrt {1 + {H^2}} }})$$. This is a generalization of Ejiri’s rigidity theorem.

On the linear Weingarten spacelike submanifolds immersed in a locally symmetric semi-Riemannian space

Let $$M^{n}$$ be an n-dimensional complete linear Weingarten spacelike submanifold immersed with parallel normalized mean curvature vector field and flat normal bundle in a locally symmetric semi-Riemannian space $$L_{p}^{n+p}$$ of index p, which obeys standard curvature constraints (such an ambient space can be regarded as an extension of a semi-Riemannian space form). In this setting, our purpose is to establish sufficient conditions guaranteeing that such a spacelike submanifold $$M^{n}$$ be either totally umbilical or isometric to an isoparametric hypersurface of a totally geodesic submanifold $$L_{1}^{n+1}\hookrightarrow L_{p}^{n+p}$$, with two distinct principal curvatures, one of which is simple. Our approach is based on a suitable Simons type formula jointly with a version of the Omori–Yau’s generalized maximum principle for a Cheng–Yau’s modified operator.

ON WEAK BIHARMONIC GENERALIZED ROTATIONAL SURFACE IN E⁴

In the present paper we consider weak biharmonic rotational surfaces in Euclidean 4-space E⁴. We have proved that the general rotational surface of parallel mean curvature vector field is weak biharmonic then either it is minimal or a constant mean curvature. Further, we show that if Vranceanu surface of constant mean curvature is weak-biharmonic then it is a Clifford torus in E⁴.

Submanifolds with parallel mean curvature vector in η-Einstein Sasaki manifold

In this note, we prove that for a 3-dimensional submanifold in an η-Einstein 5-Sasaki manifold with Einstein constant a≥4, if the Reeb vector field is tangent to the submanifold and it has parallel mean curvature form, then it is a slant submanifold. Furthermore, if a>4, then it is either an invariant submanifold or an anti-invariant submanifold.

Submanifolds with parallel Gaussian mean curvature vector in Euclidean spaces

In the present paper, we prove a rigidity theorem for complete submanifolds with parallel Gaussian mean curvature vector in the Euclidean space $${\mathbb {R}}^{n+p}$$ under an integral curvature pinching condition, which is a unified generalization of some rigidity results for self-shrinkers and the $$\lambda $$-hypersurfaces in Euclidean spaces.

A Bernstein-type theorem for -submanifolds with flat normal bundle in the Euclidean spaces

$\xi$-Submanifolds in the Euclidean spaces are a natural extension of self-shrinkers and a generalization of $\lambda$-hypersurfaces. Moreover, $\xi$-submanifolds are expected to take the place of submanifolds with parallel mean curvature vector. In this paper, we establish a Bernstein-type theorem for $\xi$-submanifolds in the Euclidean spaces. More precisely, we prove that an $n$-dimensional smooth graphic $\xi$-submanifold with flat normal bundle in $\mathbb{R}^{n+p}$ is an affine $n$-plane.

Rigidity of submanifolds in the Euclidean and spherical spaces

We deal with n-dimensional complete submanifolds $$M^n$$ immersed with nonzero parallel mean curvature vector field $$\mathbf{H}$$ either in the Euclidean space $${\mathbb {R}}^{n+p}$$ or in the Euclidean sphere $${\mathbb {S}}^{n+p}$$. In this setting, we establish sufficient conditions to guarantee that such a submanifold $$M^n$$ must be pseudo-umbilical, which means that $$\mathbf{H}$$ is an umbilical direction. Moreover, assuming a suitable lower bound for the Ricci curvature, we conclude that $$M^n$$ must be isometric to $${\mathbb {S}}^{n}$$, up to scaling.

Trapped submanifolds contained into a null hypersurface of de Sitter spacetime

We study codimension two trapped submanifolds contained into one of the two following null hypersurfaces of de Sitter spacetime: (i) the future component of the light cone, and (ii) the past infinite of the steady state space. For codimension two compact spacelike submanifolds in the light cone we show that they are conformally diffeomorphic to the round sphere. This fact enables us to deduce that the problem of characterizing compact marginally trapped submanifolds into the light cone is equivalent to solving the Yamabe problem on the round sphere, allowing us to obtain our main classification result for such submanifolds. We also fully describe the codimension two compact marginally trapped submanifolds contained into the past infinite of the steady state space and characterize those having parallel mean curvature field. Finally, we consider the more general case of codimension two complete, non-compact, weakly trapped spacelike submanifolds contained into the light cone.

Parallel mean curvature tori in $\mathbb{C}P^{2}$ and $\mathbb{C}H^{2}$

We explicitly determine tori that have a parallel mean curvature vector, both in the complex projective plane and the complex hyperbolic plane.

Timelike surfaces in Minkowski space with a canonical null direction

Given a constant vector field Z in Minkowski space, a timelike surface is said to have a canonical null direction with respect to Z if the projection of Z on the tangent space of the surface gives a lightlike vector field. For example in the three-dimensional Minkowski space: a surface has a canonical null direction if and only if it is minimal and flat. When the ambient has arbitrary dimension, if a surface has a canonical null direction and has parallel mean curvature vector then it is minimal. We give different ways for building these surfaces in the four-dimensional Minkowski space. On the other hand, we describe several properties in the non ruled general case in four-dimensional Minkowski space. One property is that the tangent part of Z is an asymptotic direction of the surface. We describe these surfaces in the ruled case in arbitrary dimension. Finally use the Gauss map for describe another properties of these surfaces in dimension four.

Complete Parallel Mean Curvature Surfaces in Two-Dimensional Complex Space-Forms

The purpose of this article is to determine explicitly the complete surfaces with parallel mean curvature vector, both in the complex projective plane and the complex hyperbolic plane. The main results are as follows: When the curvature of the ambient space is positive, there exists a unique such surface up to rigid motions of the target space. On the other hand, when the curvature of the ambient space is negative, there are `non-trivial' complete parallel mean curvature surfaces generated by Jacobi elliptic functions and they exhaust such surfaces.

Biconservative Submanifolds in $$\mathbb {S}^n\times \mathbb {R}$$ S n × R and $$\mathbb {H}^n\times \mathbb {R}$$ H n × R

In this paper we study biconservative submanifolds in \(\mathbb {S}^n\times \mathbb {R}\) and \(\mathbb {H}^n\times \mathbb {R}\) with parallel mean curvature vector field and codimesion 2. We obtain some sufficient and necessary conditions for such submanifolds to be conservative. In particular, we obtain a complete classification of 3-dimensional biconservative submanifolds in \(\mathbb {S}^4\times \mathbb {R}\) and \(\mathbb {H}^4\times \mathbb {R}\) with nonzero parallel mean curvature vector field. We also get some results for biharmonic submanifolds in \(\mathbb {S}^n\times \mathbb {R}\) and \(\mathbb {H}^n\times \mathbb {R}\).

Einstein submanifolds with parallel mean curvature

We provide a classification of Einstein submanifolds in space forms with flat normal bundle and parallel mean curvature. This extends a previous result due to Dajczer and Tojeiro (Tohoku Math J (2) 45:43–49, 1993) for isometric immersions of Riemannian manifolds with constant sectional curvature.

The generalized Lu rigidity theorem for submanifolds with parallel mean curvature

Let M be an n-dimensional oriented compact submanifold with parallel mean curvature in the unit sphere $$S^{n+p}$$ . Denote by H and S the mean curvature and the squared length of the second fundamental form of M, respectively. We obtain a classification theorem of M if it satisfies $$S+\lambda _2\le \alpha (n,H)$$ , where $$\lambda _{2}$$ is the second largest eigenvalue of the fundamental matrix and $$\alpha (n,H)$$ is defined as in Theorem B.

Biharmonic Submanifolds with Parallel Normalized Mean Curvature Vector Field in Pseudo-Riemannian Space Forms

In this paper, we investigate biharmonic submanifolds with parallel normalized mean curvature vector field in pseudo-Riemannian space forms and classify completely such pseudo-umbilical submanifolds. Also, we prove that such submanifolds have parallel mean curvature vector field under the assumption that they have diagonalizable shape operator with at most two distinct principal curvatures in the direction of the mean curvature vector field, and apply it to obtain a partial classification result.