Complete Submanifolds Research Articles

A class of complete minimal submanifolds and their associated families of genuine deformations

Concerning the problem of classifying complete submanifolds of Euclidean space with codimension two admitting genuine isometric deformations, we believe that the only known examples with the maximal possible rank four are the real Kaehler minimal submanifolds classified in parametric form by Dajczer-Gromoll [11]. They behave like minimal surfaces, namely, if simple connected they are either holomorphic or they admit a nontrivial one-parameter family of isometric deformations. Here we characterize a new class of complete genuinely deformable Euclidean submanifolds of rank four but now the structure of their second fundamental and the way it gets modified while deforming is quite more involved than in the Kaehler case. In particular, we see this as a strong indication that the classification problem is quite challenging. Being minimal, the submanifolds we introduced are also interesting by themselves. Some of the very basic question in the local and global theory of isometric immersions of Riemannian manifolds into Euclidean space remain in good part unanswered. For instance, outside some special cases it is not known which is the lowest codimension for which a given Riemannian manifold admits an isometric immersion. On one hand, there are several results that assure that a submanifold must be unique, that is, isometrically rigid, when lying in its lowest possible codimension. On the other hand, there are only a few theorems classifying isometrically deformable submanifolds and their deformations. This is due to the fact that rigidity is a “generic” property while being deformable is certainly not, and hence a situation harder to deal with. The exception for the deformation problem is the case of hypersurfaces. In fact, in the local case the problem was mostly solved by Sbrana [18] and Cartan [1] about a century ago; see [7] for details and a modern presentation. A solution to the problem for compact hypersurfaces was given by Sacksteder [17] and by Dajczer-Gromoll [10] in the complete case. But solving the deformation problem in codimension two turned out to be very challenging even in the more restrictive case of complete manifolds. In dealing with the isometric deformation problem in higher codimension, it has to be taken into account that any submanifold of a deformable submanifold has the isometric

Liouville type theorem about p-harmonic function and p-harmonic map with finite L q -energy

This paper deals with the p-harmonic function on a complete non-compact submanifold M isometrically immersed in an (n + k)-dimensional complete Riemannian manifold $$\overline M $$ of non-negative (n−1)-th Ricci curvature. The Liouville type theorem about the p-harmonic map with finite L q -energy from complete submanifold in a partially non-negatively curved manifold to non-positively curved manifold is also obtained.

P-Harmonic l-forms on Complete Noncompact Submanifolds in Sphere with Flat Normal Bundle

In this paper, we investigate a complete noncompact submanifold $$M^m$$ in a sphere $$S^{m+t}$$ with flat normal bundle. We prove that the dimension of the space of $$L^p$$ p-harmonic l-forms (when $$m\ge 4$$ , $$2\le l\le m-2$$ and when $$m=3$$ , $$l=2$$ ) on M is finite if the total curvature of M is finite and $$m\ge 3$$ . We also obtain that there are no nontrivial $$L^p$$ p-harmonic l-forms on M if the total curvature is bounded from above by a constant depending only on m, p, l.

P-Harmonic functions and connectedness at infinity of complete submanifolds in a Riemannian manifold

In this paper, we study the connectedness at infinity of complete submanifolds by using the theory of p-harmonic function. For lower-dimensional cases, we prove that if M is a complete orientable noncompact hypersurface in \(\mathbb {R}^{n+1}\) and if \(\delta \)-stability inequality holds on M, then M has only one p-nonparabolic end. It is also proved that if \(M^n\) is a complete noncompact submanifold in \({\mathbb {R}}^{n+k}\) with sufficiently small \(L^n\) norm of the traceless second fundamental form, then M has only one p-nonparabolic end. Moreover, we obtain a lower bound of the fundamental tone of the p Laplace operator on complete submanifolds in a Riemannian manifold.

Submanifolds with constant scalar curvature in a space form

We deal with complete submanifolds Mn having constant positive scalar curvature and immersed with parallel normalized mean curvature vector field in a Riemannian space form Qcn+p of constant sectional curvature c∈{1,0,−1}. In this setting, we show that such a submanifold Mn must be either totally umbilical or isometric to a Clifford torus S1(1−r2)×Sn−1(r), when c=1, a circular cylinder R×Sn−1(r), when c=0, or a hyperbolic cylinder H1(−1+r2)×Sn−1(r), when c=−1. This characterization theorem corresponds to a natural improvement of previous ones due to Alías, García-Martínez and Rigoli [2], Cheng [4] and Guo and Li [6].

Eigenvalue estimates for submanifolds with bounded f-mean curvature

In this paper, we obtain an extrinsic low bound to the first non-zero eigenvalue of the f-Laplacian on complete noncompact submanifolds of the weighted Riemannian manifold (Hm(−1),e−fdv) with respect to the f-mean curvature. In particular, our results generalize those of Cheung and Leung in Math. Z.236 (2001) 525–530.

Immersion of Kähler manifolds in the class of convex submanifolds

It is proved that a real complete convex Kahler submanifold in Euclidean space splits as a metric product of two-dimensional surfaces of positive Gaussian curvature in Euclidean 3-space and a Euclidean subspace. A theorem of V. K. Beloshapka and S. N. Bychkov is generalized to the case of convex submanifolds of any codimension.

Submanifolds with nonpositive extrinsic curvature

We prove that complete submanifolds, on which the weak Omori-Yau maximum principle for the Hessian holds, with low codimension and bounded by cylinders of small radius must have points rich in large positive extrinsic curvature. The lower the codimension is, the richer such points are. The smaller the radius is, the larger such curvatures are. This work unifies and generalizes several previous results on submanifolds with nonpositive extrinsic curvature.

L2HARMONIC 1-FORMS ON SUBMANIFOLDS WITH WEIGHTED POINCARÉ INEQUALITY

In the present note, we deal with <TEX>$L^2$</TEX> harmonic 1-forms on complete submanifolds with weighted <TEX>$Poincar{\acute{e}}$</TEX> inequality. By supposing submanifold is stable or has sufficiently small total curvature, we establish two vanishing theorems for <TEX>$L^2$</TEX> harmonic 1-forms, which are some extension of the results of Kim and Yun, Sang and Thanh, Cavalcante Mirandola and <TEX>$Vit{\acute{o}}rio$</TEX>.

Cylindricity of complete Euclidean submanifolds with relative nullity

We show that a complete Euclidean submanifold with minimal index of relative nullity \(\nu _0>0\) and Ricci curvature with a certain controlled decay must be a \(\nu _0\)-cylinder. This is an extension of the classical Hartman cylindricity theorem.

On the structure of submanifolds in the hyperbolic space

Let \(M^n\) be a complete submanifold in the hyperbolic space \(\mathbb {H}^{n+m}\). We show the vanishing of the Betti numbers \(\beta _p(M)\), \(1 \le p \le n-1\), if M is compact and the squared norm of the mean curvature satisfies some pinching condition. In the noncompact case, we prove various vanishing theorems of \(L^q\) harmonic p-forms on \(M^n\) if the mean curvature is bounded from above or below, and the total curvature is less than an explicit constant or some stability type inequality holds on \(M^n\). Finally, by putting some restrictions on the bottom of the spectrum of the Laplace operator, we can also get some vanishing theorems. On the other hand, based on the nonexistence of nontrivial \(L^2\) harmonic 1-forms on \(M^n\), we can further show some one-end theorems under various hypotheses.

On the geometry of complete submanifolds immersed in the hyperbolic space

We deal with $n$-dimensional complete submanifolds immersed with parallel nonzero mean curvature vector ${\bf H}$ in the hyperbolic space $\mathbb{H}^{n+p}$. In this setting, we establish sufficient conditions to guarantee that such a submanifold $M^n$ must be pseudo-umbilical, which means that ${\bf H}$ is an umbilical direction. In particular, we conclude that $M^n$ is a minimal submanifold of a small hypersphere of $\mathbb{H}^{n+p}$.

Isometry actions and geodesics orthogonal to submanifolds

We obtain a condition, involving geodesics orthogonal to tangent vectors, which implies that a submanifoldmust be contained in a level set of a Lipschitz function. One application is the following theorem. Let f : Σ → M be a differentiable immersion of a connected manifold Σ in a complete noncompact manifold with nonnegative sectional curvature. Fix a ray σ in M and assume that for all point p ∈ Σ and v ∈ T p Σ there exists a vector η orthogonal to df p v such that the geodesic γ η tangent to η at p is a ray asymptotic to σ. Then f(Σ) is contained in a horosphere of M associated with σ. A similar version holds in Hadamard manifolds. Another theorem studies those ideas in the context of space forms, establishing a set of equivalent conditions on a submanifold so that it is locally contained in a hypersurface invariant under the action of isometries which fix points in a given totally geodesic complete submanifold.

Vanishing theorems for [formula omitted] harmonic forms on complete submanifolds in Euclidean space

The aim of this paper is to prove vanishing theorems for L2 harmonic forms of higher order on complete non-compact submanifolds in Euclidean space. Firstly, by assuming that the submanifold has flat normal bundle, we can explicitly express the Weitzenböck formulae for harmonic p-forms. Using this formulae, we can obtain some L2 vanishing theorems by adding a relation of the square length of the second fundamental form with the squared mean curvature, or by assuming that the total curvature of the submanifold is bounded from above by an explicit positive constant. Secondly, by using a monotonicity formulae for general harmonic forms, we obtain a vanishing theorem under an appropriate decay of the norm of the second fundamental form.

Complete noncompact submanifolds with flat normal bundle

Complete noncompact submanifolds with flat normal bundle

Estimates for Eigenvalues of the Elliptic Operator in Divergence Form on Riemannian Manifolds

We investigate the Dirichlet weighted eigenvalue problem of the elliptic operator in divergence form on compact Riemannian manifolds(M,g,e-ϕdv). We establish a Yang-type inequality of this problem. We also get universal inequalities for eigenvalues of elliptic operators in divergence form on compact domains of complete submanifolds admitting special functions which include the Hadamard manifolds with Ricci curvature bounded below and any complete manifolds admitting eigenmaps to a sphere.

Convexity of Reflective Submanifolds in Special Unitary Groups

A submanifold of a Riemannian manifold is called reflective, if it is a connected component of an involutive isometry. If every shortest geodesic arc of a complete submanifold is still shortest in the ambient space, we say that the submanifold is convex. In this note we show that reflective submanifolds in special unitary groups are convex.

Vanishing theorems for [formula omitted] harmonic 1-forms on complete submanifolds in a Riemannian manifold

Let M be an n-dimensional complete orientable noncompact hypersurface in a complete Riemannian manifold of nonnegative sectional curvature. For 2≤n≤6, we prove that if M satisfies the δ-stability inequality (0<δ≤1), then there is no nontrivial L2β harmonic 1-form on M for some constant β. We also provide sufficient conditions for complete hypersurfaces to satisfy the δ-stability inequality. Moreover, we prove a vanishing theorem for L2 harmonic 1-forms on M when M is an n-dimensional complete noncompact submanifold in a complete simply-connected Riemannian manifold N with sectional curvature KN satisfying that −k2≤KN≤0 for some constant k.

On applications of Simons' type formula and reduction of codimension for complete submanifolds in space forms

We provide a survey of applications of Simons' type for- mula to submanifolds with constant mean curvature or with parallel mean curvature vector in Riemannian space forms. Also, we show a result of reduction of codimension for complete submanifolds such that the normalized mean curvature vector is parallel and the squared norm of the second fundamental form satises certain inequality. At the end, we give some open questions to submanifolds in general products of Riemannian space forms.

On complete submanifolds with parallel mean curvature in product spaces

We prove a Simons type formula for submanifolds with parallel mean curvature vector field in product spaces of type M^n(c)\times\mathbb{R} , where M^n(c) is a space form with constant sectional curvature c , and then we use it to characterize some of these submanifolds.