Complete Minimal Submanifolds Research Articles

Complete stable minimal submanifolds with a parallel unit normal section in a Euclidean space

Let Σ be an n-dimensional complete stable minimal submanifold in Rn+p admitting a parallel unit normal section ν∈Γ(NΣ). We prove that if Σ has finite L2-norm of the second fundamental form Aν with respect to ν under an additional assumption on Aη for any unit normal section η∈Γ(NΣ) such that 〈ν,η〉=0, then Σ is an affine n-plane. We also obtain an upper bound for the fundamental tone of Σ under a certain condition on the second fundamental form.

Llocp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p_{loc}$$\\end{document} Positivity Preservation and Liouville-Type Theorems

On a complete Riemannian manifold (M, g), we consider Llocp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{p}_{loc}$$\\end{document} distributional solutions of the differential inequality -Δu+λu≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-\\Delta u + \\lambda u \\ge 0$$\\end{document} with λ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda >0$$\\end{document} a locally bounded function that may decay to 0 at infinity. Under suitable growth conditions on the Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{p}$$\\end{document} norm of u over geodesic balls, we obtain that any such solution must be nonnegative. This is a kind of generalized Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{p}$$\\end{document}-preservation property that can be read as a Liouville-type property for nonnegative subsolutiuons of the equation Δu≥λu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta u \\ge \\lambda u$$\\end{document}. An application of the analytic results to Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{p}$$\\end{document} growth estimates of the extrinsic distance of complete minimal submanifolds is also given.

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.

A generalization of Pólya conjecture and Li–Yau inequalities for higher eigenvalues

We verify that if M is a compact minimal submanifold with boundary in a Cartan–Hadamard manifold N, then $$\sum \nolimits _{i=1}^{k}\lambda _i\ge \frac{2n\pi }{e}k^{\frac{n+2}{n}}{{\,\mathrm{vol}\,}}(M)^{-\frac{2}{n}}$$ , where $$\lambda _i$$ is the ith Dirichlet eigenvalue of the Laplacian on M. This provides an evidence for that the generalized Polya conjecture is true. We also prove that if M is a complete and non-compact minimal submanifold in a Cartan–Hadamard manifold N, then $$\gamma _n N_\alpha (V)\le \int _M(V-\alpha )_{-}^{n/2}(x)dx$$ , where $$N_\alpha (V)$$ is the number of eigenvalues( $$\le \alpha $$ ) of the Schrodinger operator $$-\Delta +V$$ on M and $$\gamma _n$$ is an explicit positive constant depending only on n.

Complete minimal submanifolds with nullity in Euclidean spheres

In this paper we investigate m -dimensional complete minimal submanifolds in Euclidean spheres with index of relative nullity at least m– 2 at any point. These are austere submanifolds in the sense of Harvey and Lawson [19] and were initially studied by Bryant [3]. For any dimension and codimension there is an abundance of non-complete examples fully described by Dajczer and Florit [7] in terms of a class of surfaces, called elliptic, for which the ellipse of curvature of a certain order is a circle at any point. Under the assumption of completeness, it turns out that any submanifold is either totally geodesic or has dimension three. In the latter case there are plenty of examples, even compact ones. Under the mild assumption that the Omori–Yau maximum principle holds on the manifold, a trivial condition in the compact case, we provide a complete local parametric description of the submanifolds in terms of 1-isotropic surfaces in Euclidean space. These are the minimal surfaces for which the standard ellipse of curvature is a circle at any point. For these surfaces, there exists a Weierstrass type representation that generates all simply connected ones.

Rigidity of complete minimal submanifolds in a hyperbolic space

In this paper we prove some gap theorem for complete immersed minimal submanifold of dimension no less than six or four, depending on the codimension, in a hyperbolic space $$\mathbb {H}^{n+m}(-1)$$ . That is, we show that a high dimensional complete immersed minimal submanifold M in $$ \mathbb {H}^{n+m}(-1)$$ , is totally geodesic if the $$L^d$$ norm of |A|, for some d, on geodesic balls centered at some point $$p \in M $$ has less than quadratic growth and if either $$\sup _{x \in M} |A|^2$$ is not too large or the $$L^n$$ norm of |A| on M is finite, were, A is the second fundamental form of M.

Complete Minimal Submanifolds with Nullity in the Hyperbolic Space

We investigate complete minimal submanifolds $$f: M^3\rightarrow \mathbb {H}^n$$ in hyperbolic space with index of relative nullity at least one at any point. The case when the ambient space is either the Euclidean space or the round sphere was already studied in Dajczer et al. (Math Z 287: 481–491, 2017 and Comment Math Helv, to appear, 2017), respectively. If the scalar curvature is bounded from below we conclude that the submanifold has to be either totally geodesic or a generalized cone over a complete minimal surface lying in an equidistant submanifold of $$\mathbb {H}^n$$ .

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

Complete minimal submanifolds with nullity in Euclidean space

In this paper, we investigate minimal submanifolds in Euclidean space with positive index of relative nullity. Let \(M^m\) be a complete Riemannian manifold and let \(f:M^m\rightarrow \mathbb {R}^n\) be a minimal isometric immersion with index of relative nullity at least \(m-2\) at any point. We show that if the Omori–Yau maximum principle for the Laplacian holds on \(M^m\), for instance, if the scalar curvature of \(M^m\) does not decrease to \(-\infty \) too fast or if the immersion f is proper, then the submanifold must be a cylinder over a minimal surface.

The Geometry of Grassmannian manifolds and Bernstein type theorems for higher codimension

This paper continues the line of research where one shows a Bernsteintype theorem, namely, that a complete minimal submanifold of a Euclidean space has to be affine linear, by proving that its Gauss map is constant. This Gauss map is a harmonic map into some Grassmann manifold. Therefore, we need geometric conditions on the target of the Gauss map which imply that, being harmonic, it has to be constant. For this purpose, we identify a region W13 in a Grassmann manifoldGn,m, not covered by a usualmatrix coordinate chart, with the following important property. For a complete n−submanifold M of Rn+m with n 3, and m 2, with parallel mean curvature whose image under the Gauss map is contained in a compact subset K ⇢ W13 ⇢ Gn,m, we can construct strongly subharmonic functions on M and derive a priori estimates for the harmonic Gauss map, to eventually show that it is constant. While we do not know yet how close our region is to being optimal in this respect, it is substantially larger than what could be achieved previously with other methods. Consequently, this enables us to obtain substantially stronger Bernstein-type theorems in higher codimension than previously known. We also provide some new explicit computations for the example of Lawson and Osserman [18].

Complete minimal submanifolds of compact Lie groups

We give a new method for manufacturing complete minimal submanifolds of compact Lie groups and their homogeneous quotient spaces. For this we make use of harmonic morphisms and basic representation theory of Lie groups. We then employ our method to construct many examples of compact minimal submanifolds of the special unitary groups.

Formula omitted] harmonic forms on submanifolds in a Hadamard manifold

In the present paper, we first show that an n−1n-super-stable complete minimal submanifold in Rn+m admits no nontrivial L2 harmonic 1-forms, and has only one end, which generalized Cao–Shen–Zhu’s result on stable minimal hypersurface in Rn+1. For L2 harmonic forms of higher order, we prove a vanishing and finiteness theorem under the assumptions on Schrödinger operators involving the squared norm of the traceless second fundamental form. Finally, if the potential function of the Schrödinger operator is a multiple of the squared norm of the second fundamental form, we can further show that the hypersurface has only one end or finitely many ends.

Optimal isoperimetric inequalities for complete proper minimal submanifolds in hyperbolic space

Abstract Let Σ be a k-dimensional complete proper minimal submanifold in the Poincaré ball model Bn of hyperbolic geometry. If we consider Σ as a subset of the unit ball Bn in Euclidean space, we can measure the Euclidean volumes of the given minimal submanifold Σ and the ideal boundary ∂ ∞ Σ $\partial _\infty \Sigma $ , say Vol ℝ ( Σ ) $\operatorname{Vol}_{\mathbb {R}}(\Sigma )$ and Vol ℝ ( ∂ ∞ Σ ) $\operatorname{Vol}_{\mathbb {R}}(\partial _\infty \Sigma )$ , respectively. Using this concept, we prove an optimal linear isoperimetric inequality. We also prove that if Vol ℝ ( ∂ ∞ Σ ) ≥ Vol ℝ ( 𝕊 k - 1 ) $\operatorname{Vol}_{\mathbb {R}}(\partial _\infty \Sigma ) \ge \operatorname{Vol}_{\mathbb {R}}(\mathbb {S}^{k-1})$ , then Σ satisfies the classical isoperimetric inequality. By proving the monotonicity theorem for such Σ, we further obtain a sharp lower bound for the Euclidean volume Vol ℝ ( Σ ) $\operatorname{Vol}_{\mathbb {R}}(\Sigma )$ , which is an extension of Fraser–Schoen and Brendle's recent results to hyperbolic space. Moreover we introduce the Möbius volume of Σ in Bn to prove an isoperimetric inequality via the Möbius volume for Σ.

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.

EIGENVALUE ESTIMATES FOR QUADRATIC POLYNOMIAL OPERATOR OF THE LAPLACIAN

AbstractFor a bounded domain Ω in a complete Riemannian manifold M, we investigate the Dirichlet weighted eigenvalue problem of quadratic polynomial operator Δ2 − aΔ + b of the Laplacian Δ, where a and b are the nonnegative constants. We obtain an inequality for eigenvalues which contains a constant that only depends on the mean curvature of M. It yields an upper bound of the (k + 1)th eigenvalue Λk + 1. As their applications, some inequalities and bounds of eigenvalues on a complete minimal submanifold in a Euclidean space and a unit sphere are obtained.

Universal bounds for eigenvalues of the biharmonic operator

In this paper, we study the eigenvalues of the clamped plate problem: { Δ 2 u = λ u , in D , u | ∂ D = ∂ u ∂ ν | ∂ D = 0 , where D is a bounded connected domain in an n-dimensional complete minimal submanifold of a unit m-sphere S m ( 1 ) or of an m-dimensional Euclidean space R m . Let 0 < λ 1 < λ 2 ⩽ ⋯ ⩽ λ k ⩽ ⋯ be the eigenvalues of the above problem. We obtain universal bounds on λ k + 1 in terms the first k eigenvalues independent of the domains. For example, when D is contained in an n-dimensional complete minimal submanifold of S m ( 1 ) , we show that λ k + 1 − 1 k ∑ i = 1 k λ i ⩽ 1 k n { ∑ i = 1 k ( λ k + 1 − λ i ) 1 / 2 ( ( 2 n + 4 ) λ i 1 / 2 + n 2 ) } 1 / 2 ⋅ { ∑ i = 1 k ( λ k + 1 − λ i ) 1 / 2 ( 4 λ i 1 / 2 + n 2 ) } 1 / 2 , from which one can obtain a more explicit upper bound on λ k + 1 in terms of λ 1 , … , λ k (see Corollary 1). When D is contained in a complete n-dimensional minimal submanifold of R m , we prove the inequality λ k + 1 ⩽ 1 k ∑ i = 1 k λ k + ( 8 ( n + 2 ) n 2 ) 1 / 2 1 k ∑ i = 1 k ( λ i ( λ k + 1 − λ i ) ) 1 / 2 which generalizes the main theorem in Cheng and Yang (2006) [10] that states that the same estimate holds when D is a connected and bounded domain in R n .

RIGIDITY OF MINIMAL SUBMANIFOLDS WITH FLAT NORMAL BUNDLE

Abstract. Let M n be a complete immersed super stable minimal sub-manifold in R n + p with °at normal bundle. We prove that if M has flnitetotal L 2 norm of its second fundamental form, then M is an a–ne n -plane. We also prove that any complete immersed super stable minimalsubmanifold with °at normal bundle has only one end. 1. IntroductionLet M be an n -dimensional complete minimal submanifold in R n + p . When n = 2 and p = 1, do Carmo and Peng [3], Fischer-Colbrie and Schoen [5]independently showed that the only complete stable minimal surface is a plane.Recall that a minimal submanifold is stable if the second variation of its volumeis always nonnegative for any normal variation with compact support. Forhigher dimensional minimal hypersurfaces, do Carmo and Peng [4] generalizedthe result mentioned as above. We will denote by A the second fundamentalform of M .Theorem ([4]). Let M n be a complete stable minimal hypersurface in R n +1 satisfying R M jAj 2 dv < 1. Then M must be a hyperplane.

Minimal submanifolds with small total scalar curvature in Euclidean space

Let M be an n-dimensional complete minimal submanifold in Rn+p. Lei Ni proved that if M has sufficiently small total scalar curvature, then M has only one end. We improve the upper bound of total scalar curvature. We also prove that if M has the same upper bound of total scalar curvature, there is no nontrivial L2 harmonic 1-form on M.

Hamiltonian minimal submanifolds in complex hyperbolic space with [formula omitted]-symmetry

In this paper, we construct complete Hamiltonian minimal submanifolds in complex hyperbolic space with SO ( n ) -symmetry which are not minimal in the usual sense.

The embedded singly periodic Scherk-Costa surfaces

We give a positive answer to M. Traizet's open question about the existence of complete embedded minimal surfaces with Scherk-ends without planar geodesics. In the singly periodic case, these examples get close to an extension of Traizet's result concerning asymmetric complete minimal submanifolds of Open image in new window with finite total curvature.