Complete Submanifolds Research Articles

A simple improvement of a differentiable classification result for complete submanifolds

We consider $M^n$, $n\geq3$, an $n$-dimensional complete submanifold of a Riemannian manifold $(\overline{M}^{n+p},\overline{g})$. We prove that if for all point $x\in M^n$ the following inequality is satisfied $$S\leq\frac{8}{3} \bigg( \overline{K}_{\min}-\frac{1}{4}\overline{K}_{\max} \bigg)+\frac{n^2H^2}{n-1},$$ with strictly inequality at one point, where $S$ and $H$ denote the squared norm of the second fundamental form and the mean curvature of $M^n$ respectively, then $M^n$ is either diffeomorphic to a spherical space form or the Euclidean space $\mathbb{R}^n$. In particular, if $M^n$ is simply connected, then $M^n$ is either diffeomorphic to the sphere $\mathbb{S}^n$ or the Euclidean space $\mathbb{R}^n$.

Rigidity theorems for complete spacelike submanifold in $$S^{n+p}_q(1)$$

In this paper, by investigating complete spacelike submanifold with \(R={ aH}+b\) in semi-Riemannian manifold \(S^{n+p}_q(1)\,(1\le q\le p)\), two rigidity theorems are obtained, which are a generalization of main results in Chaves and Sousa (Differ Geom Appl 25:419–432, 2007) and Han and Feng (Bull Math Anal Appl 3:97–108, 2011).

Volume Growth, Number of Ends, and the Topology of a Complete Submanifold

Given a complete isometric immersion φ:Pm⟶Nn in an ambient Riemannian manifold Nn with a pole and with radial sectional curvatures bounded from above by the corresponding radial sectional curvatures of a radially symmetric space \(M^{n}_{w}\), we determine a set of conditions on the extrinsic curvatures of P that guarantee that the immersion is proper and that P has finite topology in line with the results reported in Bessa et al. (Commun. Anal. Geom. 15(4):725–732, 2007) and Bessa and Costa (Glasg. Math. J. 51:669–680, 2009). When the ambient manifold is a radially symmetric space, an inequality is shown between the (extrinsic) volume growth of a complete and minimal submanifold and its number of ends, which generalizes the classical inequality stated in Anderson (Preprint IHES, 1984) for complete and minimal submanifolds in ℝn. As a corollary we obtain the corresponding inequality between the (extrinsic) volume growth and the number of ends of a complete and minimal submanifold in hyperbolic space, together with Bernstein-type results for such submanifolds in Euclidean and hyperbolic spaces, in the manner of the work Kasue and Sugahara (Osaka J. Math. 24:679–704, 1987).

Geometric and differentiable rigidity of submanifolds in spheres

In this paper, we investigate rigidity of geometric and differentiable structures of complete submanifolds via an extrinsic geometrical quantity τ(x) defined by the second fundamental form. We verify a geometric rigidity theorem for complete submanifolds with parallel mean curvature in a unit sphere Sn+p. Inspired by the rigidity theorem, we prove a differentiable sphere theorem for complete submanifolds in Sn+p. Moreover, we obtain a differentiable pinching theorem for complete submanifolds in a δ(>14)-pinched Riemannian manifold.

The Gauss image of entire graphs of higher codimension and Bernstein type theorems

Under suitable conditions on the range of the Gauss map of a complete submanifold of Euclidean space with parallel mean curvature, we construct a strongly subharmonic function and derive a-priori estimates for the harmonic Gauss map. The required conditions here are more general than in previous work and they therefore enable us to improve substantially previous results for the Lawson–Osseman problem concerning the regularity of minimal submanifolds in higher codimension and to derive Bernstein type results.

Convex function on pseudo-Grassmann manifold and its applications for Bernstein-type theorem

We derive the Hessian estimate of the w-function defined on the pseudo-Grassmann manifold Gn,mm, which is convex by the estimate. As an application, we give a direct proof of the Bernstein-type theorem due to Y. Xin [Manuscripta Math., 2000, 103: 191–202]. We also estimate the squared norm of the second fundamental form of a complete spacelike submanifold in pseudo-Euclidean space in terms of the w-function and the mean curvature.

Some Bernstein-type rigidity theorems

We prove some Bernstein-type rigidity theorems for complete submanifolds in a Euclidean space and space-like submanifolds of a Lorentzian space. In particular, we obtain a Bernstein rigidity theorem for complete minimal submanifolds of arbitrary codimension in Euclidean space.

Bernstein type theorems for complete submanifolds in space forms

AbstractWe study the Bernstein type problem for complete submanifolds in the space forms. In particular, we prove that any complete super stable minimal submanifolds in an (n + p)‐dimensional Euclidean space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^{n+p}\,(n\le 5)$\end{document} with finite L1 norm of the second fundamental form must be affine n‐dimensional planes. We also prove that any complete noncompact weakly stable hypersurfaces in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^{n+1}\,(n\le 5)$\end{document} with constant mean curvature and finite Ld (d = 1, 2, 3) norm of traceless second fundamental form must be hyperplanes.

Splitting Theorems for Submanifolds of Nonnegative Isotropic Curvature

We study isometric immersions of complete manifolds of nonnegative isotropic curvature that have spaces of relative nullity. These manifolds decompose into a Riemannian product $${\bar{M}\times \mathbb R^k}$$ . In the case that k ≥ 2, we use recent results of Brendle-Schoen to completely classify these manifolds. We then apply this classification to study complete non-compact Kahler submanifolds with relatively low codimension.

On complete submanifolds with bounded mean curvature

We consider M n , n ≥ 3 , an n -dimensional complete and connected submanifold of a space form M n + p ( c ) , whose mean curvature H does not vanish and is bounded with a parallel normalized mean curvature vector. We prove that if S ≤ n 2 H 2 n − 1 + 2 c , where S denotes the squared norm of the second fundamental form of M n , then the codimension of isometric immersion reduces to 1. This result generalizes the case where the mean curvature vector is parallel or mean curvature is constant. If M n is a compact submanifold of hyperbolic space form with constant mean curvature H ≠ 0 , then M n is a geodesic sphere. When M n is a submanifold of the unit sphere with constant mean curvature H ≠ 0 , then either M n is a great or small sphere in S n + 1 ( 1 ) or M n is a product of spheres S l ( r ) × S m ( s ) .

Good Shadows, Dynamics and Convex Hulls of Complete Submanifolds

Good Shadows, Dynamics and Convex Hulls of Complete Submanifolds

Complete submanifolds of manifolds of negative curvature

In this article, we give sufficient conditions for a complete non-compact submanifold in a simply connected manifold of negative sectional curvature to have only one end. We also prove some Liouville theorems for harmonic maps from the same kind of submanifolds to manifolds of non-positive curvature.

Inequalities for eigenvalues of elliptic operators in divergence form on Riemannian manifolds

In this paper, we study eigenvalues of elliptic operators in divergence form on compact Riemannian manifolds with boundary (possibly empty) and obtain a general inequality for them. By using this inequality, we prove universal inequalities for eigenvalues of elliptic operators in divergence form on compact domains of complete submanifolds in a Euclidean space, and of complete manifolds admitting special functions which include the Hadamard manifolds with Ricci curvature bounded below, a class of warped product manifolds, the product of Euclidean spaces with any complete manifold and manifolds admitting eigenmaps to a sphere.

L2 harmonic 1-forms on complete submanifolds in Euclidean space

Let Mn (n ≥ 3) be an n-dimensional complete noncompact oriented submanifold in an (n+p)-dimensional Euclidean space Rn+p with finite total mean curvature, i.e, ∫M|H|n < ∞, where H is the mean curvature vector of M. Then we prove that each end of M must be non-parabolic. Denote by φ the traceless second fundamental form of M. We also prove that if ∫M|φ|n < C (n), where C (n) is an an explicit positive constant, then there are no nontrivial L2 harmonic 1-forms on M and the first de Rham's cohomology group with compact support of M is trivial. As corollaries, such a submanifold has only one end. This implies that such a minimal submanifold is plane.

Complete submanifolds with bounded mean curvature in a Hadamard manifold

Let Q be an m -dimensional Hadamard manifold and let M be an n -dimensional complete non-compact submanifold in Q . Let κ be a non-positive constant and suppose that the ( n − 1 ) -th Ricci curvature of Q is no less than ( n − 1 ) κ . Denote by Δ , A and H the Laplacian operator, the second fundamental form and the mean curvature vector of M , respectively. Assume that | H | ≤ H 0 < ∞ and when κ = 0 we also assume that | H | ≥ H 1 > 0 . In the present paper, we prove finiteness theorems on ends of M using the theory of L 2 harmonic 1 -forms. In particular, we show that if n ≥ 3 and if the index of the operator L ≡ Δ − ( n − 1 ) κ + n − 1 2 | A | 2 is finite, then M has finitely many ends; if n ≥ 2 and if the index of L is zero, then M has only one end.

Complete submanifolds in manifolds of partially non-negative curvature

We prove that a complete non-compact submanifold in a complete manifold of partially non-negative sectional curvature has only one end if the Sobolev inequality holds on it and if its total curvature is not very big by showing a Liouville theorem for harmonic maps and by using a existence theorem of constant harmonic functions with finite energy. We also generalize a result by Cao–Shen–Zhu saying that a complete orientable stable minimal hypersurface in a Euclidean space has only one end to submanifolds in manifolds of partially non-negative sectional curvature. Some related results about the structure of the same kind of submanifolds are also obtained.

ON SUBMANIFOLDS WITH TAMED SECOND FUNDAMENTAL FORM

AbstractBased on the ideas of Bessa, Jorge and Montenegro (Comm. Anal. Geom., vol. 15, no. 4, 2007, pp. 725–732) we show that a complete submanifold M with tamed second fundamental form in a complete Riemannian manifold N with sectional curvature KN ≤ κ ≤ 0 is proper (compact if N is compact). In addition, if N is Hadamard, then M has finite topology. We also show that the fundamental tone is an obstruction for a Riemannian manifold to be realised as submanifold with tamed second fundamental form of a Hadamard manifold with sectional curvature bounded below.

Total curvature and L 2 harmonic 1-forms on complete submanifolds in space forms

Let Mn be an n-dimensional complete noncompact oriented submanifold with finite total curvature, i.e., \({\int_M(|A|^2-n|H|^2)^{\frac n2} < \infty}\), in an (n + p)-dimensional simply connected space form Nn+p(c) of constant curvature c, where |H| and |A|2 are the mean curvature and the squared length of the second fundamental form of M, respectively. We prove that if M satisfies one of the following: (i) n ≥ 3, c = 0 and \({\int_M|H|^n < \infty}\); (ii) n ≥ 5, c = −1 and \({|H| < 1-\frac{2}{\sqrt n}}\); (iii) n ≥ 3, c = 1 and |H| is bounded, then the dimension of the space of L2 harmonic 1-forms on M is finite. Moreover, in the case of (i) or (ii), M must have finitely many ends.

Topological and differentiable sphere theorems for complete submanifolds

Topological and differentiable sphere theorems for complete submanifolds

VANISHING AND TOPOLOGICAL SPHERE THEOREMS FOR SUBMANIFOLDS IN A HYPERBOLIC SPACE

We extend the vanishing and sphere theorems due to Lawson, Simons, Xin, Shiohama and Xu. By using the techniques of calculus of variations in the geometric measure theory, we prove the vanishing theorem for homology groups of submanifolds in the hyperbolic space Hn(c) with negative constant curvature c. Moreover, we obtain a topological sphere theorem for certain complete submanifolds in Hn(c).