Real hypersurfaces of a complex space form

In this paper we are interested in obtaining a condition under which a compact real hypersurface of a complex projective space CP ⁿ is a geodesic sphere. We also study the question as to whether the characteristic vector field of a real hypersurface of the complex projective space CP ⁿ is harmonic, and show that the answer is in negative.     

The Hamiltonian dynamics over real hypersurfaces in Kaehler manifolds

Let X be a Kaehler manifold with complex dimension n. Let ωX be its Kaehler form. Let M be a strongly pseudo convex real hypersurface in X. For this hypersurface, the deformation theory of CR structures is successfully developed. And we find that H¹(M,T^′) (the T^′-valued Kohn-Rossi cohomology) is the Zariski tangent space of the versal family. In this paper, the geometrical meaning of H¹(M,O) is studied, and we propose to study displacements of the real hypersurface, which preserves the type of the differential form, ωX, over CR structures, on M, infinitesimally.     

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.     

A characterization of a certain real hypersurface of type (A<Subscript>2</Subscript>) in a complex projective space

In the class of real hypersurfaces M ^2n?1 isometrically immersed into a nonflat complex space form \(\widetilde {{M_n}}\left( c \right)\) of constant holomorphic sectional curvature c (≠ 0) which is either a complex projective space ?P ⁿ (c) or a complex hyperbolic space ?H ⁿ (c) according as c > 0 or c < 0, there are two typical examples. One is the class of all real hypersurfaces of type (A) and the other is the class of all ruled real hypersurfaces. Note that the former example are Hopf manifolds and the latter are non-Hopf manifolds. In this paper, inspired by a simple characterization of all ruled real hypersurfaces in \(\widetilde {{M_n}}\left( c \right)\), we consider a certain real hypersurface of type (A₂) in ?P ⁿ (c) and give a geometric characterization of this Hopf manifold.     

First nonzero eigenvalue of a pseudo-umbilical hypersurface in the unit sphere

S. Deshmukh has obtained interesting results for first nonzero eigenvalue of a minimal hypersurface in the unit sphere. In the present article, we generalize these results to pseudoumbilical hypersurface and prove: What conditions are satisfied by the first nonzero eigenvalue λ ₁ of the Laplacian operator on a compact immersed pseudo-umbilical hypersurface M in the unit sphere S ⁿ⁺¹. We also show that a compact immersed pseudo-umbilical hypersurface of the unit sphere S ⁿ⁺¹ with λ ₁ = n is either isometric to the sphere S ⁿ or for this hypersurface an inequaluity is fulfilled in which sectional curvatures of the hypersuface M participate.     

Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster \mathfrak{D}^ \bot-parallel structure Jacobi operator

Regarding the generalized Tanaka-Webster connection, we considered a new notion of \(\mathfrak{D}^ \bot\) -parallel structure Jacobi operator for a real hypersurface in a complex two-plane Grassmannian G ₂(? ^m+2) and proved that a real hypersurface in G ₂(? ^m+2) with generalized Tanaka-Webster \(\mathfrak{D}^ \bot\) -parallel structure Jacobi operator is locally congruent to an open part of a tube around a totally geodesic quaternionic projective space ?P ⁿ in G ₂(? ^m+2), where m = 2n.     

Möbius Isoparametric Hypersurfaces in S n+1 with Two Distinct Principal Curvatures

A hypersurface x : M → S ⁿ⁺¹ without umbilic point is called a Möbius isoparametric hypersurface if its Möbius form Φ = ?ρ ^?2∑ _i (e _i (H) + ∑ _j (h _ij ?Hδ _ij )e _j (log ρ))θ _i vanishes and its Möbius shape operator $ {\Bbb {S}}A hypersurface x : M → S ^{n +1} without umbilic point is called a M?bius isoparametric hypersurface if its M?bius form Φ = −ρ⁻²∑ _i (e _i (H) + ∑ _j (h _ij −Hδ _ij )e _j (log ρ))θ _i vanishes and its M?bius shape operator ? = ρ⁻¹(S−Hid) has constant eigenvalues. Here {e _i } is a local orthonormal basis for I = dx·dx with dual basis {θ _i }, II = ∑ _ij h _ij θ _i ⊗θ _i is the second fundamental form, and S is the shape operator of x. It is clear that any conformal image of a (Euclidean) isoparametric hypersurface in S ^{n +1} is a M?bius isoparametric hypersurface, but the converse is not true. In this paper we classify all M?bius isoparametric hypersurfaces in S ^{n +1} with two distinct principal curvatures up to M?bius transformations. By using a theorem of Thorbergsson [1] we also show that the number of distinct principal curvatures of a compact M?bius isoparametric hypersurface embedded in S ^{n +1} can take only the values 2, 3, 4, 6. Received September 7, 2001, Accepted January 30, 2002     

Normal connections induced by a framed hypersurface in the conformal space

We find two normal connections induced by the normal framing of a hypersurface V _{n ? 1} in the conformal space C _n , and establish relationship between these connections and a Weyl connection which is also induced by the normal framing of V _{n ? 1}. We study two normal connections induced by a complete framing of a hypersurface V _{n ? 1} in C _n . We establish relationship between geometries of a framed hypersurface V _{n ? 1} of the conformal space C _n and a quadratic hyperband of the projective space P _{n + 1} associated with V _{n ? 1}.     

Complete Spacelike Submanifolds in de Sitter Space with Parallel Mean Curvature Vector Satisfying H2 = 4(n − 1)/n2

Recently, Montiel [7] proved that an n-dimensional (n 3) complete spacelike hypersurface in de Sitter space S^{1 1 +1}(1) with constant mean curvature H satisfying H² = 4 (n – 1)/n² which is not connected at infinity must be, up to rigidity motion, a certain hyperbolic cylinder. In this paper, we prove that Montiel's result still holds for higher codimensional spacelike submanifolds in de Sitter space Sⁿ _p ^+p(1).     

On some finite ramified coverings of Pn

Let S be a hypersurface in Pⁿ (n≧3) with only normal crossings and let ƒ : XPⁿ be a finite ramified covering which is unramified over Pⁿ − S. Then S. Kawai has shown that there are neither regular 1-forms nor regular 2-forms on X. The aim of this article is to derive a stronger conclusion: H⁰(X,Ω_X^p)= 0 for 1≦p<n , and moreover H⁰(X,Ω_X^p)= 0 if deg S≦n+1.     

Smooth hypersurfaces with a CR contraction

In this paper we show that a C^∞ real hypersurface in Cn+1 of finite D'Angelo type admitting a weakly contracting local CR automorphism is CR equivalent to a weighted homogeneous hypersurface. As an application, we show that a bounded pseudoconvex domain in Cn+1 with C^∞ boundary of finite D'Angelo type with a hyperbolic orbit accumulation point is biholomorphically equivalent to a domain defined by a weighted homogeneous polynomial.     

Dual Riemannian spaces of constant curvature on a normalized hypersurface

In this paper, the following results are obtained: 1) It is proved that, in the fourth order differential neighborhood, a regular hypersurface V _n−1 embedded into a projective-metric space K _n , n ≥ 3, intrinsically induces a dual projective-metric space $ \bar K_n $ \bar K_n . 2) An invariant analytical condition is established under which a normalization of a hypersurface V _n−1 ⊂ K _n (a tangential hypersurface $ \bar V_{n - 1} $ \bar V_{n - 1} ⊂ $ \bar K_n $ \bar K_n ) by quasitensor fields H _n ⁱ , H _i ($ \bar H_n^i $ \bar H_n^i , $ \bar H_i $ \bar H_i ) induces a Riemannian space of constant curvature. If the two conditions are fulfilled simultaneously, the spaces R _n−1 and $ \bar R_{n - 1} $ \bar R_{n - 1} are spaces of the same constant curvature $ K = - \tfrac{1} {c} $ K = - \tfrac{1} {c} . 3) Geometric interpretations of the obtained analytical conditions are given.     

A characterization of totally <Emphasis Type="Italic">η</Emphasis>-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form

We give a characterization of totally η-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form in terms of totally umbilical condition for the holomorphic distribution on real hypersurfaces. We prove that if the shape operator A of a real hypersurface M of a complex space form M ⁿ (c), c ≠ 0, n ⩾ 3, satisfies g(AX, Y) = ag(X, Y) for any X, Y ∈ T ₀(x), a being a function, where T ₀ is the holomorphic distribution on M, then M is a totally η-umbilical real hypersurface or locally congruent to a ruled real hypersurface. This condition for the shape operator is a generalization of the notion of η-umbilical real hypersurfaces.     

Geometric, topological and differentiable rigidity of submanifolds in space forms

Let M be an n-dimensional submanifold in the simply connected space form F ^n+p (c) with c + H ² > 0, where H is the mean curvature of M. We verify that if M ⁿ (n ≥ 3) is an oriented compact submanifold with parallel mean curvature and its Ricci curvature satisfies Ric _M ≥ (n ? 2)(c + H ²), then M is either a totally umbilic sphere, a Clifford hypersurface in an (n + 1)-sphere with n = even, or ${\mathbb{C}P^{2} \left(\frac{4}{3}(c + H^{2})\right) {\rm in} S^{7} \left(\frac{1}{\sqrt{c + H^{2}}}\right)}$ C P 2 4 3 ( c + H 2 ) in S 7 1 c + H 2 . In particular, if Ric _M > (n ? 2)(c + H ²), then M is a totally umbilic sphere. We then prove that if M ⁿ (n ≥ 4) is a compact submanifold in F ^n+p (c) with c ≥ 0, and if Ric _M > (n ? 2)(c + H ²), then M is homeomorphic to a sphere. It should be emphasized that our pinching conditions above are sharp. Finally, we obtain a differentiable sphere theorem for submanifolds with positive Ricci curvature.     

The Willmore functional and the containment problem in R4

Let Σ be a convex hypersurface in the Euclidean space R ⁴ with mean curvature H. We obtain a geometric lower bound for the Willmore functional ∫_Σ H ² dσ. This bound is an invariant involving the area of Σ, the volume and Minkowski quermassintegrals of the convex body that Σ bounds. We also obtain a sufficient condition for a convex body to contain another in the Euclidean space R ⁴.     

Rigidity theorems for hypersurfaces with constant mean curvature

Let M ⁿ be a compact oriented hypersurface of a unit sphere \(\mathbb{S}^{n + 1} \) (1) with constant mean curvature H. Given an integer k between 2 and n ? 1, we introduce a tensor ? related to H and to the second fundamental form A of M, and show that if |?|² ≤ B _H,k and tr(? ³) ≤ C _n,k |?|³, where B _H,k and C _n,k are numbers depending only on H, n and k, then either |?|² ≡ 0 or |?|² ≡ B _H,k . We characterize all M ⁿ with |?|² ≡ B _H,k . We also prove that if \(\left| A \right|^2 \leqslant 2\sqrt {k(n - k)}\) and tr(? ³) ≤ C _n,k |?|³ then |A|² is constant and characterize all M ⁿ with |A|² in the interval \(\left[ {0,2\sqrt {k\left( {n - k} \right)} } \right] \) . We also study the behavior of |?|², with the condition additional tr(? ³) ≤ C _n,k |?|³, for complete hypersurfaces with constant mean curvature immersed in space forms and show that if sup _M |?|² = B _H,k and this supremum is attained in M ⁿ then M ⁿ is an isoparametric hypersurface with two distinct principal curvatures of multiplicities k y n ? k. Finally, we use rotation hypersurfaces to show that the condition on the trace of ? ³ is necessary in our results; more precisely, for each integer k with 2 ≤ k ≤ n ? 1 and \(H \geqslant 1/\sqrt {2n - 1} \) there is a complete hypersurface M ⁿ in \(\mathbb{S}^{n + 1} \) (1) with constant mean curvature H such that sup _M |?|² = B _H,k , and this supremum is attained in M ⁿ , and which is not a product of spheres.     

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).     

Classification of polynomial translation hypersurfaces of finite type

In this paper we prove that any hypersurface in Eⁿ⁺¹ of the form where P ₁ is a polynomial of degree ≥2 and P ₂, ... , P _n are functions such that P _i ^′ P _i ^″ = 0 somewhere for all i = 2, ... , n, is of infinite type. As a consequence, we deduce that a polynomial translation hypersurface in Eⁿ⁺¹, i. e. a hypersurface of the above form where P ₁, ... , P _n are polynomials, is of finite type if and only if it is a hyperplane. This provides some partial solutions to a problem of B. Y. Chen [C3].     

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     

Internal geometry of hypersurfaces in projectively metric space

In this paper, we study the internal geometry of a hypersurface V _n−1 embedded in a projectively metric space K _n , n ≥ 3, and equipped with fields of geometric-objects { G_nⁱ,G_i } \left\{ {G_n^i,{G_i}} \right\} and { H_nⁱ,G_i } \left\{ {H_n^i,{G_i}} \right\} in the sense of Norden and with a field of a geometric object { H_nⁱ,H_n } \left\{ {H_n^i,{H_n}} \right\} in the sense of Cartan. For example, we have proved that the projective-connection space P _n−1,n−1 induced by the equipment of the hypersurface V_{n - 1}   ì   K_n,  n 3 3 {V_{n - 1}}\; \subset \;{K_n},\;n \geq 3 , in the sense of Cartan with the field of a geometrical object { H_nⁱ,H_n } \left\{ {H_n^i,{H_n}} \right\} is flat if and only if its normalization by the field of the object { H_nⁱ,G_i } \left\{ {H_n^i,{G_i}} \right\} in the tangent bundle induces a Riemannian space R _n−1 of constant curvature K = −1/c.