Complete Submanifolds Research Articles

Sharp pinching theorems for complete submanifolds in the sphere

Abstract For every complete and minimally immersed submanifold f : M n → S n + p f\colon M^{n}\to\mathbb{S}^{n+p} whose second fundamental form satisfies | A | 2 ≤ n ⁢ p / ( 2 ⁢ p − 1 ) \lvert A\rvert^{2}\leq np/(2p-1) , we prove that it is either totally geodesic, or (a covering of) a Clifford torus or a Veronese surface in S 4 \mathbb{S}^{4} , thereby extending the well-known results by Simons, Lawson and Chern, do Carmo & Kobayashi from compact to complete M n M^{n} . We also obtain the corresponding result for complete hypersurfaces with non-vanishing constant mean curvature, due to Alencar & do Carmo in the compact case, under the optimal bound on the umbilicity tensor. In dimension n ≤ 6 n\leq 6 , a pinching theorem for complete higher-codimensional submanifolds with non-vanishing parallel mean curvature is proved, partly generalizing previous work by Santos. Our approach is inspired by the conformal method of Fischer-Colbrie, Shen & Ye and Catino, Mastrolia & Roncoroni.

Vanishing Theorems for p-Harmonic Forms on Submanifolds in Spheres

In this paper, we present some vanishing theorems for p-harmonic forms on -super stable complete submanifold M immersed in sphere Sn + m. When 2 ≤ 1 ≤ n-2, M has a flat normal bundle. Assuming that M is a minimal submanifold and δ > 1(n–1)p2 / 4n[p–1+(p–1)2kp], we prove a vanishing theorem for p-harmonic ℓ-forms.

On complete submanifolds with parallel normalized mean curvature in product spaces

A Simons type formula for submanifolds with parallel normalized mean curvature vector field (pnmc submanifolds) in the product spaces $M^{n}(c)\times \mathbb {R}$ , where $M^{n}(c)$ is a space form with constant sectional curvature $c\in \{-1,1\}$ , it is shown. As an application is obtained rigidity results for submanifolds with constant second mean curvature.

Stochastically complete submanifolds with parallel mean curvature vector field in a Riemannian space form

Stochastically complete submanifolds with parallel mean curvature vector field in a Riemannian space form

Submanifolds immersed in Riemannian spaces endowed with a Killing vector field: Nonexistence and rigidity

Assuming suitable constraints on the warping function ρ and on the mean curvature vector field, we apply the Omori-Yau's generalized maximum principle in order to study the behavior of a support function naturally attached to a complete n-dimensional submanifold Σn immersed in a Killing warped product Mn+p×ρR whose base Mn+p has sectional curvature bounded from below. In the case that Σn is compact, we prove the nonexistence of non-minimal compact mean curvature flow solitons with respect to a non-singular Killing vector field. When Σn has codimension 2, we conclude that there do not exist n-dimensional minimal compact submanifolds immersed in Mn+1×ρR whose base is either Mn+1=Rn+1 or Mn+1=Hn+1. We also get that the only minimal topological 2-spheres immersed in S3×ρR are the totally geodesic ones in a slice S3×{t0}. Moreover, we obtain further results concerning the geometry of complete submanifolds immersed in a Killing warped product, under appropriate restrictions on the Bakry-Émery-Ricci tensor, Ricψ, of these submanifolds with respect to the weight function ψ=ln⁡ρ−2.

Submanifolds immersed in a warped product with density

We study $n$-dimensional complete submanifolds immersed in a weighted warped product of the type $I\times_fM^{n+p}_{\varphi}$, whose warping function $f$ has convex logarithm and weight function $\varphi$ does not depend on the real parameter $t\in I$. Assuming the constancy of an appropriate support function involving the $\varphi$-mean curvature vector field of such a submanifold $\Sigma^n$ jointly with suitable constraints on the Bakry-Emery-Ricci tensor of $\Sigma^n$, we prove that it must be contained in a slice of the ambient space. As applications, we obtain codimension reductions and Bernstein-type results for complete $\varphi$-minimal bounded multi graphs constructed over the $n$-dimensional Gaussian space. Our approach is based on the weak Omori-Yau's generalized maximum principle and Liouville-type results for the drift Laplacian.

Rigidity of Complete Minimal Submanifolds in Spheres

Let $M$ be an $n$-dimensional complete minimal submanifold in an $(n+p)$-dimensional sphere $S^{n+p}$ and $h$ be the second fundamental form of $M$. In this paper, it is shown that $M$ is totally geodesic if the $L^2$ norm of $|h|$ on any geodesic ball of $M$ has less than quadratic growth and $L^n$ norm of $|h|$ on $M$ is less than a fixed constant. Farther, deleting the condition of the $L^2$ norm of $|h|$, we show that $M$ is totally geodesic if $L^n$ norm of $|h|$ on $M$ is less than a fixed constant. Furthermore, we provide a sufficient condition such that a complete stable minimal hypersurface is totally geodesic.

Complete submanifolds with relative nullity in space forms

We use techniques based on the splitting tensor to explicitly integrate the Codazzi equation along the relative nullity distribution and express the second fundamental form in terms of the Jacobi tensor of the ambient space. This approach allows us to easily recover several important results in the literature on complete submanifolds with relative nullity of the sphere as well as derive new strong consequences in hyperbolic and Euclidean spaces. Among the consequences of our main theorem are results on submanifolds with sufficiently high index of relative nullity, submanifolds with nonpositive extrinsic curvature and submanifolds with integrable relative conullity. We show that no complete submanifold of hyperbolic space with sufficiently high index of relative nullity has extrinsic geometry bounded away from zero. As an application of these results, we derive an interesting corollary for complete submanifolds of hyperbolic space with nonpositive extrinsic curvature and discourse on their relation to Milnor’s conjecture about complete surfaces with second fundamental form bounded away from zero. Finally, we also prove that every complete Euclidean submanifold with integrable relative conullity is a cylinder over the relative conullity.

A gap theorem for complete submanifolds with parallel mean curvature in the hyperbolic space

Let M be an n(≥7)-dimensional complete submanifold with parallel mean curvature in the hyperbolic space Hn+p, whose mean curvature satisfies H2−1≤0. Denote by A˚ and BR(q) the trace free second fundamental form of M and the geodesic ball of radius R centered at q∈M, respectively. We prove that if lim supR→∞∫BR(q)|A˚|2dMR2=0, and if (∫M|A˚|ndM)2/n+2n(n−2)3n(n−1)H(∫M|A˚|n/2dM)2/n≤C(n), then M is congruent to an n-dimensional hyperbolic space or the Euclidean space Rn. Here C(n) is an explicit positive constant depending only on n. We also obtain a similar gap theorem in the case where n=5,6.

Geometric Inequalities of Warped Product Submanifolds and Their Applications

In the present paper, we prove that if Laplacian for the warping function of complete warped product submanifold M m = B p × h F q in a unit sphere S m + k satisfies some extrinsic inequalities depending on the dimensions of the base B p and fiber F q such that the base B p is minimal, then M m must be diffeomorphic to a unit sphere S m . Moreover, we give some geometrical classification in terms of Euler–Lagrange equation and Hamiltonian of the warped function. We also discuss some related results.

Complete spacelike submanifolds in a locally symmetric semi-Riemannian space

In this paper, complete spacelike submanifolds with parallel normalized mean curvature vector are investigated in semi-Riemannian space obeying some standard curvature conditions. In this setting, we obtain a suitable Simons type formula and apply it jointly with the well-known generalized maximum principle of Omori–Yau to show that it must be totally umbilical submanifold or isometric to an isoparametric hypersurface in a submanifold [Formula: see text] of [Formula: see text].

The p-eigenvalue estimates and Lqp-harmonic forms on submanifolds of Hadamard manifolds

In this paper, we first shows a lower bound estimate for the first eigenvalue of the p-Laplacian on a complete submanifold in a simply connected manifold of negative sectional curvature, in terms of Ln-norm of the mean curvature. In the second part, we prove some vanishing theorems of Lqp-harmonic ℓ-forms on complete noncompact submanifolds in Hadamard manifolds, by assuming the index of certain Schrödinger operators are zero, or by assuming integral or pointwise pinching conditions on the mean curvature and the traceless second fundamental form.

$$L^p$$ harmonic 1-forms on totally real submanifolds in a complex projective space

Let $$\pi : {\mathbb {S}}^{2n+1}\rightarrow {\mathbb {C}}P^n$$ be the Hopf map and let $$\phi$$ be a totally real immersion of a $$k(\ge 3)$$-dimensional simply connected manifold $$\Sigma$$ into $${\mathbb {C}}P^n$$. It is well known that there exists an isotropic lift $${\overline{\phi }}$$ into $${\mathbb {S}}^{2n+1}$$ preserving the second fundamental form. Using this isotropic lift, we obtain a vanishing theorem for of $$L^{p}$$ harmonic 1-forms on a complete noncompact totally real submanifold in a complex projective space provided the $$L^k$$ norm of the traceless second fundamental form $$\Phi$$ is sufficiently small. Moreover, we prove that if the $$L^k$$ norm of $$\Phi$$ is finite, then the dimension of $$L^p$$ harmonic 1-forms on a complete noncompact totally real submanifold in a complex projective space is finite. As consequences, we obtain a vanishing theorem and a finiteness result for $$L^2$$ harmonic 1-forms on a complete noncompact minimal Lagrangian submanifold in a complex projective space.

Volume Growth of Complete Submanifolds in Gradient Ricci Solitons with Bounded Weighted Mean Curvature

Abstract In this article, we study properly immersed complete noncompact submanifolds in a complete shrinking gradient Ricci soliton with weighted mean curvature vector bounded in norm. We prove that such a submanifold must have polynomial volume growth under some mild assumption on the potential function. On the other hand, if the ambient manifold is of bounded geometry, we prove that such a submanifold must have at least linear volume growth. In particular, we show that a properly immersed complete noncompact hypersurface in the Euclidean space with bounded Gaussian weighted mean curvature must have polynomial volume growth and at least linear volume growth.

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

In this paper, we study eigenvalue of linear fourth order elliptic operators in divergence form with Dirichlet boundary condition on a bounded domain in a compact Riemannian manifolds with boundary (possibly empty) and provide a general inequality for them. As an application, by using this inequality, we study eigenvalues of this operator on compact domains of complete submanifolds in a Euclidean space.

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.

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.

Vanishing Theorem for p-Harmonic 1-Forms on Complete Submanifolds in Spheres

In this paper, we study a complete submanifold $$M^m$$ in a sphere $$S^{m+l}$$ . We obtain that there is no nontrivial $$L^{2\beta }$$ p-harmonic 1-forms on $$M^m$$ if the total curvature is bounded from above by a constant depending only on m, and we also obtain that $$M^m$$ has only one p-nonparabolic end.

On the fundamental tone of immersions and submersions

In this paper we obtain lower bound estimates of the spectrum of the Laplace-Beltrami operator on complete submanifolds with bounded mean curvature, whose ambient space admits a Riemannian submersion over a Riemannian manifold with negative sectional curvature. Our main theorem generalizes many previously known estimates and applies for both immersions and submersions.

English

We deal with complete submanifolds with weighted Poincare inequality. By assuming the submanifold is δ-stable or has sufficiently small total curvature, we establish two vanishing theorems for L p harmonic 1-forms, which are extensions of the results of Dung-Seo and Cavalcante-Mirandola-Vitorio.