Hypersurfaces In Complex Space Research Articles

Real hypersurfaces in complex space forms with special almost contact structures

<abstract><p>In this paper, we prove that an almost contact metric structure of a real hypersurface in a complex space form is quasi-contact if and only if it is contact. We also classify real hypersurfaces whose associated almost contact metric structures are nearly Kenmotsu or cosymplectic, which gives several extensions of some earlier results in this field.</p></abstract>

The equivalence theory for infinite type hypersurfaces in ℂ²

We develop a classification theory for real-analytic hypersurfaces in C 2 \mathbb {C}^{2} in the case when the hypersurface is of infinite type at the reference point. This is the remaining, not yet understood case in C 2 \mathbb {C}^{2} in the Problème local, formulated by H. Poincaré in 1907 and asking for a complete biholomorphic classification of real hypersurfaces in complex space. One novel aspect of our results is a notion of smooth normal forms for real-analytic hypersurfaces. We rely fundamentally on the recently developed CR-DS technique in CR-geometry.

Quasi-Einstein Hypersurfaces of Complex Space Forms

Based on a well-known fact that there are no Einstein hypersurfaces in a nonflat complex space form, in this article, we study the quasi-Einstein condition, which is a generalization of an Einstein metric, on the real hypersurface of a nonflat complex space form. For the real hypersurface with quasi-Einstein metric of a complex Euclidean space, we also give a classification. Since a gradient Ricci soliton is a special quasi-Einstein metric, our results improve some conclusions of Cho and Kimura.

On the Chern–Moser–Weyl tensor of real hypersurfaces

We derive an explicit formula for the well-known Chern–Moser–Weyl tensor for nondegenerate real hypersurfaces in complex space in terms of their defining functions. The formula is considerably simplified when applying to “pluriharmonic perturbations” of the sphere or to a Fefferman approximate solution to the complex Monge–Ampère equation. As an application, we show that the CR invariant one-form $X_{\alpha}$ constructed recently by Case and Gover is nontrivial on each real ellipsoid of revolution in $\mathbb{C}^3$, unless it is equivalent to the sphere. This resolves affirmatively a question posed by these two authors in 2017 regarding the (non-) local CR invariance of the $\mathcal{I}'$-pseudohermitian invariant in dimension five and provides a counterexample to a recent conjecture by Hirachi.

On the Minkowski Formula for Hypersurfaces in Complex Space Forms

Abstract In this paper, we discuss various Minkowski-type formulas for real hypersurfaces in complex space forms. In particular, we investigate the formulas suggested by the natural splitting of the tangent space. In this direction, our main result concerns a new kind of 2nd Minkowski formula.

Generalized Sasakian Space Forms Which Are Realized as Real Hypersurfaces in Complex Space Forms

We prove a classification theorem of the generalized Sasakian space forms which are realized as real hypersurfaces in complex space forms.

Commuting structure Jacobi operators for real hypersurfaces in complex space forms II

Let $M$ be a real hypersurface in a complex space form $M_n(c)$, $c \not= 0$. In this paper, we prove that if the structure Jacobi operator $R_\xi$ is $\phi\nabla_{\xi}\xi$-parallel and $R_\xi$ commute with the Ricci tensor, then $M$ is a Hopf hypersurface provided that the mean curvature of $M$ is constant with respect to the structure vector field.

Ricci Curvature of Real Hypersurfaces in Non-flat Complex Space Forms

We establish an inequality among the Ricci curvature, the squared mean curvature, and the normal curvature for real hypersurfaces in complex space forms. We classify real hypersurfaces in two-dimensional non-flat complex space forms which admit a unit vector field satisfying identically the equality case of the inequality.

On the embeddability of real hypersurfaces into hyperquadrics

A well known result of Forstnerić [15] states that most real-analytic strictly pseudoconvex hypersurfaces in complex space are not holomorphically embeddable into spheres of higher dimension. A more recent result by Forstnerić [16] states even more: most real-analytic hypersurfaces do not admit a holomorphic embedding even into a merely algebraic hypersurface of higher dimension, in particular, a hyperquadric. We emphasize that both cited theorems are proved by showing that the set of embeddable hypersurfaces is a set of first Baire category. At the same time, the classical theorem of Webster [30] asserts that every real-algebraic Levi-nondegenerate hypersurface admits a transverse holomorphic embedding into a nondegenerate real hyperquadric in complex space.In this paper, we provide effective results on the non-embeddability of real-analytic hypersurfaces into a hyperquadric. We show that, under the codimension restriction N≤2n, the defining functions φ(z,z¯,u) of all real-analytic hypersurfaces M={v=φ(z,z¯,u)}⊂Cn+1 containing Levi-nondegenerate points and locally transversally holomorphically embeddable into some hyperquadric Q⊂CN+1 satisfy an universal algebraic partial differential equation D(φ)=0, where the algebraic-differential operator D=D(n,N) depends on n≥1, n<N≤2n only. To the best of our knowledge, this is the first effective result characterizing real-analytic hypersurfaces embeddable into a hyperquadric of higher dimension. As an application, we show that for every n,N as above there exists μ=μ(n,N) such that a Zariski generic real-analytic hypersurface M⊂Cn+1 of degree ≥μ is not transversally holomorphically embeddable into any hyperquadric Q⊂CN+1. We also provide an explicit upper bound for μ in terms of n,N. To the best of our knowledge, this gives the first effective lower bound for the CR-complexity of a Zariski generic real-algebraic hypersurface in complex space of a fixed degree.

Some integral formulas for the characteristic curvature

ABSTRACTWe show some integral formulas involving the characteristic curvature for closed real hypersurfaces in complex spaces.

Non-Hopf Hypersurfaces in 2-dimensional Complex Space Forms

In this paper we give a geometric characterization of non-Hopf hypersurfaces in the complex space form $M^2(c)$ under a condition on the shape operator. We also classify pseudo-parallel real hypersurfaces of $M^2(c)$.

ON THE LIE DERIVATIVE OF REAL HYPERSURFACES IN ℂP2AND ℂH2WITH RESPECT TO THE GENERALIZED TANAKA-WEBSTER CONNECTION

Abstract. In this paper the notion of Lie derivative of a tensor fieldT of type (1,1) of real hypersurfaces in complex space forms with re-spect to the generalized Tanaka-Webster connection is introduced andis called generalized Tanaka-Webster Lie derivative. Furthermore, threedimensional real hypersurfaces in non-flat complex space forms whosegeneralized Tanaka-Webster Lie derivative of 1) shape operator, 2) struc-ture Jacobi operator coincides with the covariant derivative of them withrespect to any vector field X orthogonal to ξ are studied. 1. IntroductionA complex space form is an n-dimensional Kahler manifold of constant holo-morphic sectional curvature c. A complete and simply connected complex spaceform is analytically isometric to a complex projective space CP n if c > 0, or toa complex Euclidean space C n if c = 0, or to a complex hyperbolic space CH n if c < 0. The complex projective and complex hyperbolic spaces are callednon-flat complex space forms, since c 6= 0 and the symbol M

On complete list of affine homogeneous surfaces of (ɛ, 0)-types in the space ℂ3

The problem of description of affine homogeneous real hypersurfaces in complex spaces is an important part of the problem of holomorphic classification of homogeneous manifolds, which has no complete solution till now, even in the 3-dimensional case. The method developed by the authors, based on affine canonical equations and techniques of matrix Lie algebras, allowed them earlier to obtain complete descriptions of two natural classes of affine homogeneous real hypersurfaces in the 3-dimensional complex space. In this paper, we present the complete description of one more class. The description includes examples already known and (obtained with the use of symbolic computations) Lie algebras corresponding to the other homogeneous manifolds belonging to the types under consideration. The main result of the paper is obtained by means of integration of these algebras.

On Classification of Real Hypersurfaces in a Complex Space Form with $\eta$-recurrent Shape Operator

In this paper, we classify real hypersurfaces in a non-flat complex space with η-recurrent shape operator.

SOBOLEV ESTIMATES FOR THE LOCAL EXTENSION OF BOUNDARY HOLOMORPHIC FORMS ON REAL HYPERSURFACES IN ℂn

Let M be a smooth real hypersurface in complex space of dimension <TEX>$n$</TEX>, <TEX>$n{\geq}3$</TEX>, and assume that the Levi-form at <TEX>$z_0$</TEX> on M has at least <TEX>$(q+1)$</TEX>-positive eigenvalues, <TEX>$1{\leq}q{\leq}n-2$</TEX>. We estimate solutions of the local <TEX>$\bar{\partial}$</TEX>-closed extension problem near <TEX>$z_0$</TEX> for <TEX>$(p,q)$</TEX>-forms in Sobolev spaces. Using this result, we estimate the local solution of tangential Cauchy-Riemann equation near <TEX>$z_0$</TEX> in Sobolev spaces.

Commuting Structure Jacobi Operator for Real Hypersurfaces in Complex Space Forms

Let M be a real hypersurface of a complex space form with almost contact metric structure (φ,ξ,η,g). In this paper, we prove that if the structure Jacobi operator Rξ=(·,ξ) ξ is φ▽ξξ-parallel and Rξ commute with the shape operator, then M is a Hopf hypersurface. Further, if Rξ is φ▽ξξ-parallel and Rξ commute with the Ricci tensor, then M is also a Hopf hypersurface provided that TrRξ is constant.

Transversality of holomorphic mappings between real hypersurfaces in complex spaces of different dimensions

We consider holomorphic mappings $H$ between a smooth real hypersurface $M\subset \bC^{n+1}$ and another $M'\subset \bC^{N+1}$ with $N\geq n$. We provide conditions guaranteeing that $H$ is transversal to $M'$ along all of $M$. In the strictly pseudoconvex case, this is well known and follows from the classical Hopf boundary lemma. In the equidimensional case ($N=n$), transversality holds for maps of full generic rank provided that the source is of finite type in view of recent results by the authors (see also a previous paper by the first author and L. Rothschild). In the positive codimensional case ($N>n$), the situation is more delicate as examples readily show. In recent work by S. Baouendi, the first author, and L. Rothschild, conditions were given guaranteeing that the map $H$ is transversal outside a proper subvariety of $M$, and examples were given showing that transversality may fail at certain points. One of the results in this paper implies that if $N\le 2n-2$, $M'$ is Levi-nondegenerate, and $H$ has maximal rank outside a complex subvariety of codimension 2, then $H$ is transversal to $M'$ at all points of $M$. We show by examples that this conclusion fails in general if $N\geq 2n$, or if the set $W_H$ of points where $H$ is not of maximal rank has codimension one. We also show that $H$ is transversal at all points if $H$ is assumed to be a finite map (which allows $W_H$ to have codimension one) and the stronger inequality $N\leq 2n-3$ holds, provided that $M$ is of finite type.

Sobolev estimates for the local extension of ∂̄_{𝑏}-closed (0,1)-forms on real hypersurfaces in ℂⁿ with two positive eigenvalues

Let M \mathcal M be a smooth real hypersurface in complex space of dimension n ≥ 3 n\ge 3 , and assume that the Levi-form at z 0 z_0 on M \mathcal M has at least two positive eigenvalues. We estimate solutions of the local ∂ ¯ \bar {\partial } -closed extension problem near z 0 z_0 for ( 0 , 1 ) (0,1) -forms in Sobolev spaces. Using this result, we estimate the local solution of tangential Cauchy-Riemann equations near z 0 z_0 for ( 0 , 1 ) (0,1) -forms in Sobolev spaces.

Non-Hopf real hypersurfaces with constant principal curvatures in complex space forms

We classify real hypersurfaces in complex space forms with constant principal curvatures and whose Hopf vector field has two nontrivial projections onto the principal curvature spaces. In complex projective spaces such real hypersurfaces do not exist. In complex hyperbolic spaces these are holomorphically congruent to open parts of tubes around the ruled minimal submanifolds with totally real normal bundle introduced by Berndt and Bruck. In particular, they are open parts of homogenous ones. © Indiana University Mathematics Journal.

On Three Dimensional Real Hypersurfaces in Complex Space Forms

On Three Dimensional Real Hypersurfaces in Complex Space Forms