Non-degenerate Real Hypersurfaces Research Articles

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.

Homogeneous Levi non-degenerate hypersurfaces in {{mathbb {C}}}^3

We classify all (locally) homogeneous Levi non-degenerate real hypersurfaces in {{mathbb {C}}}^3 with symmetry algebra of dimension ge 6.

Hopf Real Hypersurfaces in the Indefinite Complex Projective Space

We wish to attack the problems that H.~Anciaux and K.~Panagiotidou posed in [1], for non-degenerate real hypersurfaces in indefinite complex projective space. We will slightly change these authors' point of view, obtaining cleaner equations for the almost contact metric structure. To make the theory meaningful, we construct new families of non-degenerate Hopf real hypersurfaces whose shape operator is diagonalisable, and one Hopf example with degenerate metric and non-diagonalisable shape operator. Next, we obtain a rigidity result. We classify those real hypersurfaces which are $\eta$-umbilical. As a consequence, we characterize some of our new examples as those whose Reeb vector field $\xi$ is Killing.

REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS WITH GENERALIZED TANAKA–WEBSTER 𝔇⊥-PARALLEL SHAPE OPERATOR

In a paper due to [I. Jeong, H. Lee and Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with generalized Tanaka–Webster parallel shape operator, Kodai Math. J.34 (2011) 352–366] we have shown that there does not exist a hypersurface in G2(ℂm+2) with parallel shape operator in the generalized Tanaka–Webster connection (see [N. Tanaka, On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections, Japan J. Math.20 (1976) 131–190; S. Tanno, Variational problems on contact Riemannian manifolds, Trans. Amer. Math. Soc.314(1) (1989) 349–379]). In this paper, we introduce a new notion of generalized Tanaka–Webster 𝔇⊥-parallel for a hypersurface M in G2(ℂm+2), and give a characterization for a tube around a totally geodesic ℍ Pnin G2(ℂm+2) where m = 2n.

Explicit bounds for the finite jet determination problem

We introduce biholomorphic invariants for (germs of) rigid holomorphically nondegenerate real hypersurfaces in complex space and show how they can be used to compute explicit bounds on the order of jets for which biholomorphisms of the hypersurface are determined uniquely by their jets. The main result which allows us to derive these bounds is a theorem which shows that solutions of certain singular analytic equations are uniquely determined by their 1 1 -jet.

On the Inhomogeneous Yang-Mills Equation dD*R D=f

We build a canonical family {Ds } of Hermitian connections in a Hermitian CR-holomorphic vector bundle (E,h ) over a nondegenerate CR manifold M, parametrized by S ∈ Γ ∞(End (E )), S skewsymmetric. Consequently, we prove an existence and uniqueness result for the solution to the inhomogeneous Yang-Mills equation dD*R D =f on M. As an application we solve for D ∈D (E,h ) when E is either the trivial line bundle, or a locally trivial CR-holomorphic vector bundle over a nondegenerate real hypersurface in a complex manifold, or a canonical bundle over a pseudo-Einstein CR manifold.

On holomorphic vector fields on compact non-degenerate real hypersurfaces of complex manifolds

On holomorphic vector fields on compact non-degenerate real hypersurfaces of complex manifolds

Non-degenerate real hypersurfaces in complex manifolds admitting large groups of pseudo-conformal transformations II

This is the continuation of our previous paper [3], and will complete, without homogeneity assumption, the classification of non-degenerate real hypersurfaces S of complex manifolds M for which the groups A(S) of pseudo-conformal transformations of S have either the largest dimension n2 + 2n or the second largest dimension.

Non-degenerate real hypersurfaces in complex manifolds admitting large groups of pseudo-conformal transformations. I

Let S (resp. S′) be a (real) hypersurface (i.e. a real analytic sub-manifold of codimension 1) of an n-dimensional complex manifold M (resp. M′). A homeomorphism f of S onto S′ is called a pseudo-conformal homeomorphism if it can be extended to a holomorphic homeomorphism of a neighborhood of S in M onto a neighborhood of S′ in M. In case such an f exists, we say that S and S′ are pseudo-conformally equivalent. A hypersurface S is called non-degenerate (index r) if its Levi-form is non-degenerate (and its index is equal to r) at each point of S.

On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections

On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections