Real hypersurfaces in the complex hyperbolic quadrics with isometric Reeb flow

We classify real hypersurfaces with isometric Reeb flow in the complex hyperbolic quadrics [Formula: see text], [Formula: see text]. We show that [Formula: see text] is even, say [Formula: see text], and any such hypersurface becomes an open part of a tube around a [Formula: see text]-dimensional complex hyperbolic space [Formula: see text] which is embedded canonically in [Formula: see text] as a totally geodesic complex submanifold or a horosphere whose center at infinity is [Formula: see text]-isotropic singular. As a consequence of the result, we get the nonexistence of real hypersurfaces with isometric Reeb flow in odd-dimensional complex quadrics [Formula: see text], [Formula: see text].

REAL HYPERSURFACES WITH ISOMETRIC REEB FLOW IN COMPLEX QUADRICS

We classify real hypersurfaces with isometric Reeb flow in the complex quadrics Qm = SOm+2/SOmSO2, m ≥ 3. We show that m is even, say m = 2k, and any such hypersurface is an open part of a tube around a k-dimensional complex projective space ℂPk which is embedded canonically in Q2k as a totally geodesic complex submanifold. As a consequence, we get the non-existence of real hypersurfaces with isometric Reeb flow in odd-dimensional complex quadrics Q2k+1, k ≥ 1. To our knowledge the odd-dimensional complex quadrics are the first examples of homogeneous Kähler manifolds which do not admit a real hypersurface with isometric Reeb flow.

SEMIGLOBAL EXTENSION OF MAXIMALLY COMPLEX SUBMANIFOLDS

Let A be a domain of the boundary of a (weakly) pseudoconvex domain Ω of ℂn and M a smooth, closed, maximally complex submanifold of A. We find a subdomain E of ℂn, depending only on Ω and A, and a complex variety W⊂E such that bW=M in E. Moreover, a generalization to analytic sets of depth at least 4 is given.

On the differentiability of a distance function

Let M be a simply connected complete K?hler manifold and N a closed complete totally geodesic complex submanifold of M such that every minimal geodesic in N is minimal in M. Let U? be the unit normal bundle of N in M. We prove that if a distance function ? is differentiable at v ? U?, then ? is also differentiable at -v.

CR-SUBMANIFOLDS OF A QUASl-KAEHLER MANIFOLD

Let $M$ be a CR-submanifold of a quasi-Kaehler manifold $N$. Sufficient conditions for the holomorphic distribution $D$ in $M$ to be integrable are derived. We also show that $D$ is minimal. It follows that an (almost) complex submanifold of a quasi-Kaehler manifold is minimal, this generalizes the well known result that a complex submanifold of a Kaehler manifold is minimal.