First nonzero eigenvalue of a minimal hypersurface in the unit sphere

In this paper, it is shown that the first nonzero eigenvalue λ₁ of the Laplacian operator on a compact immersed minimal hypersurface M in the unit sphere S ⁿ⁺¹ satisfies one of the following $$ (i)\lambda _{1}=n, \quad (ii)\lambda _{1} \leq (1+k_{0})n, \quad (iii)\lambda _{1}\geq n+\frac{n}{2}(nk_{0}-(n-1))$$ where k ₀ is the infimum of the sectional curvatures of M. It is also shown that a compact immersed minimal hypersurface of the unit sphere S ⁿ⁺¹ with λ₁?=?n is either isometric to the unit sphere S ⁿ or else k ₀?<?n ^?1(n?1).     

Eigenvalue estimates for minimal hypersurfaces in hyperbolic space

Recently Candel [A. Candel, Eigenvalue estimates for minimal surfaces in hyperbolic space, Trans. Amer. Math. Soc. 359 (2007) 3567-3575] proved that if M is a simply-connected stable minimal surface isometrically immersed in H³, then the first eigenvalue of M satisfies 1/4?λ(M)?4/3 and he asked whether the bound is sharp and gave an example such that the lower bound is attained. In this note, we prove that the upper bound can never be attained. Also we extend the result by proving that if M is compact stable minimal hypersurface isometrically immersed in Hn+1 where n?3 such that its smooth Yamabe invariant is negative, then (n−1)/4?λ(M)?n²(n−2)/(7n−6).     

Classification of hypersurfaces with parallel Möbius second fundamental form in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis>+1</Superscript>

Let M ⁿ(n ≥ 2) be an immersed umbilic-free hypersurface in the (n+1)-dimensional unit sphere S ⁿ⁺¹. Then M ⁿ is associated with a so-called Möbius metric g, and a Möbius second fundamental form B which are invariants of M ⁿunder the Möbius transformation group of S ⁿ⁺¹. In this paper, we classify all umbilic-free hypersurfaces with parallel Möbius second fundamental form.     

A first eigenvalue estimate for embedded hypersurfaces

Suppose that M is a compact orientable hypersurface embedded in a compact n-dimensional orientable Riemannian manifold N. Suppose that the Ricci curvature of N is bounded below by a positive constant k. We show that 2λ₁>k−(n−1)maxM|H| where λ₁ is the first eigenvalue of the Laplacian of M and H is the mean curvature of M.     

A new characterization of Willmore submanifolds

Let M ⁿ be a compact Willmore submanifold in the unit sphere S ^n+p . In this note, we investigate the first eigenvalue of the Schrödinger operator L = ?Δ?q on M, where q is some potential function on M, and present a gap estimate for the first eigenvalue of L.     

Small eigenvalues of large Hermitian moment matrices

We consider an infinite Hermitian positive definite matrix M which is the moment matrix associated with a measure μ with infinite and compact support on the complex plane. We prove that if the polynomials are dense in L²(μ) then the smallest eigenvalue λn of the truncated matrix Mn of M of size (n+1)×(n+1) tends to zero when n tends to infinity. In the case of measures in the closed unit disk we obtain some related results.     

A characterization of Moebius isoparametric hypersurfaces of the sphere

We consider a proper, umbilic-free immersion of an n-dimensional manifold M in the sphere S ⁿ⁺¹. We show that M is a Moebius isoparametric hypersurface if, and only if, it is a cyclide of Dupin or a Dupin hypersurface with constant Moebius curvature.     

Immersed Hypersurfaces in the Unit Sphere S^m＋1 with Constant Blaschke Eigenvalues

For an immersed submanifold x ： M^m→ Sn in the unit sphere S^n without umbilics, an eigenvalue of the Blaschke tensor of x is called a Blaschke eigenvalue of x. It is interesting to determine all hypersurfaces in Sn with constant Blaschke eigenvalues. In this paper, we are able to classify all immersed hypersurfaces in S^m＋1 with vanishing MSbius form and constant Blaschke eigenvalues, in case （1） x has exact two distinct Blaschke eigenvalues, or （2） m = 3. With these classifications, some interesting examples are also presented.     

Constant mean curvature hypersurfaces with two principal curvatures in a sphere

In this paper we consider a compact oriented hypersurface M ⁿ with constant mean curvature H and two distinct principal curvatures λ and μ with multiplicities (n − m) and m, respectively, immersed in the unit sphere S ⁿ⁺¹. Denote by f_ij{\phi_{ij}} the trace free part of the second fundamental form of M ⁿ , and Φ be the square of the length of f_ij{\phi_{ij}} . We obtain two integral formulas by using Φ and the polynomial P_H,m(x)=x²+ \fracn(n-2m)?{nm(n-m)}H x -n(1+H²){P_{H,m}(x)=x^{2}+ \frac{n(n-2m)}{\sqrt{nm(n-m)}}H x -n(1+H^{2})} . Assume that B _H,m is the square of the positive root of P _H,m (x) = 0. We show that if M ⁿ is a compact oriented hypersurface immersed in the sphere S ⁿ⁺¹ with constant mean curvatures H having two distinct principal curvatures λ and μ then either F = B_H,m{\Phi=B_{H,m}} or F = B_H,n-m{\Phi=B_{H,n-m}} . In particular, M ⁿ is the hypersurface S^n-m(r)×S^m(?{1-r²}){S^{n-m}(r)\times S^{m}(\sqrt{1-r^{2}})} .     

Spherical submanifolds in a Euclidean space

In this paper, we are interested in extending the study of spherical curves in R ³ to the submanifolds in the Euclidean space R ^n+p . More precisely, we are interested in obtaining conditions under which an n-dimensional compact submanifold M of a Euclidean space R ^n+p lies on the hypersphere S ^n+p−1(c) (standardly imbedded sphere in R ^n+p of constant curvature c). As a by-product we also get an estimate on the first nonzero eigenvalue of the Laplacian operator Δ of the submanifold (cf. Theorem 3.5) as well as a characterization for an n-dimensional sphere S ⁿ (c) (cf. Theorem 4.1).     

A characterization of Moebius isoparametric hypersurfaces of the sphere

We consider a proper, umbilic-free immersion of an n-dimensional manifold M in the sphere S ⁿ⁺¹. We show that M is a Moebius isoparametric hypersurface if, and only if, it is a cyclide of Dupin or a Dupin hypersurface with constant Moebius curvature.     

The rigidity of Clifford torusS^1 \left( {\sqrt {\frac{1}{n}} } \right) \times S^{n - 1} \left( {\sqrt {\frac{{n - 1}}{n}} } \right)

In this paper, we prove that ifM is ann-dimensional closed minimal hypersurface with two distinct principal curvatures of a unit sphereS ⁿ⁺¹ (1), thenS=n andM is a Clifford torus ifn≤S≤n+[2n ²(n+4)/3(n(n+4)+4)], whereS is the squared norm of the second fundamental form ofM.     

Compact minimal hypersurfaces with index one in the real projective space

Let M ⁿ be a compact (two-sided) minimal hypersurface in a Riemannian manifold . It is a simple fact that if has positive Ricci curvature then M cannot be stable (i.e. its Jacobi operator L has index at least one). If is the unit sphere and L has index one, then it is known that M must be a totally geodesic equator.?We prove that if is the real projective space , obtained as a metric quotient of the unit sphere, and the Jacobi operator of M has index one, then M is either a totally geodesic sphere or the quotient to the projective space of the hypersurface obtained as the product of two spheres of dimensions n ₁, n ₂ and radius R ₁, R ₂, with and . Received: June 6, 1998     

Existence and Topological Uniqueness of Compact CMC Hypersurfaces with Boundary in Hyperbolic Space

It is proved that if Γ is a compact, embedded hypersurface in a totally geodesic hypersurface ? ⁿ of ? ⁿ⁺¹ satisfying the enclosing H-hypersphere condition with |H|<1, then there is one and only one (up to a reflection on ? ⁿ ) compact embedded constant mean curvature H hypersurface M such that ?M=Γ. Moreover, M is diffeomorphic to a ball.     

Stability index jump for constant mean curvature hypersurfaces of spheres

It is known that the totally umbilical hypersurfaces in the (n + 1)-dimensional spheres are characterized as the only hypersurfaces with weak stability index 0. That is, a compact hypersurface with constant mean curvature, cmc, in S ⁿ⁺¹, different from an Euclidean sphere, must have stability index greater than or equal to 1. In this paper we prove that the weak stability index of any non-totally umbilical compact hypersurface ${M \subset S^{{n+1}}}$ with cmc cannot take the values 1, 2, 3 . . . , n.     

Spacelike hypersurfaces with constant scalar curvature

In this paper, we shall give an integral equality by applying the operator □ introduced by S.Y. Cheng and S.T. Yau [7] to compact spacelike hypersurfaces which are immersed in de Sitter space S ^{n +1} ₁(c) and have constant scalar curvature. By making use of this integral equality, we show that such a hypersurface with constant scalar curvature n(n-1)r is isometric to a sphere if r << c. Received: 18 December 1996 / Revised version: 26 November 1997     

Spacelike hypersurfaces with constant scalar curvature

In this paper, we shall give an integral equality by applying the operator □ introduced by S.Y. Cheng and S.T. Yau [7] to compact spacelike hypersurfaces which are immersed in de Sitter spaceS ₁ ⁿ⁺¹ (c) and have constant scalar curvature. By making use of this integral equality, we show that such a hypersurface with constant scalar curvaturen(n−1)r is isometric to a sphere ifr<c. Research partially Supported by a Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Science and Culture.     

Eigenvalue pinching and application to the stability and the almost umbilicity of hypersurfaces

In this paper we give pinching theorems for the first nonzero eigenvalue of the Laplacian on compact hypersurfaces of ambient spaces with bounded sectional curvature. As an application we deduce a rigidity result for stable constant mean curvature hypersurfaces M of these spaces N. Indeed, we prove that if M is included in a ball of radius small enough then the Hausdorff-distance between M and a geodesic sphere S of N is small. Moreover M is diffeomorphic and quasi-isometric to S. As other application, we obtain rigidity results for almost umbilic hypersurfaces.     

Constant mean curvature hypersurfaces with two principal curvatures in a sphere

In this paper we consider a compact oriented hypersurface M ⁿ with constant mean curvature H and two distinct principal curvatures λ and μ with multiplicities (n − m) and m, respectively, immersed in the unit sphere S ⁿ⁺¹. Denote by the trace free part of the second fundamental form of M ⁿ , and Φ be the square of the length of . We obtain two integral formulas by using Φ and the polynomial . Assume that B _H,m is the square of the positive root of P _H,m (x) = 0. We show that if M ⁿ is a compact oriented hypersurface immersed in the sphere S ⁿ⁺¹ with constant mean curvatures H having two distinct principal curvatures λ and μ then either or . In particular, M ⁿ is the hypersurface .      

A sharp height estimate for compact spacelike hypersurfaces with constant r-mean curvature in the Lorentz-Minkowski space and application

In this paper we obtain a sharp height estimate concerning compact spacelike hypersurfaces Σn immersed in the (n+1)-dimensional Lorentz-Minkowski space Ln+1 with some nonzero constant r-mean curvature, and whose boundary is contained into a spacelike hyperplane of Ln+1. Furthermore, we apply our estimate to describe the nature of the end of a complete spacelike hypersurface of Ln+1.