Cauchy-Riemann Submanifold Research Articles

Extension of positive current across small subsets

The purpose of this paper is to prove an extension result through a closed subset A having small Hausdorff dimension for a positive plurisubharmonic (Psh) current T of dimension p such that ddcT is of locally finite mass. As a consequence we prove that, if there exists a positive (resp. negative) current S such that T = ddcS on Ω\A, then the trivial extension Ŧ of T is a closed positive current in the casewhere A is a Cauchy-Riemann submanifold of class C1 and of dimension CR equal to p − 1.

<i>C</i><sup>∞</sup>-REGULARITY OF THE TANGENTIAL CAUCHY-RIEMANN EQUATIONS ON LEVI-FLAT SUBMANIFOLDS OF C<sup><i>n</sup></i>

Regularity of solutions of the tangential Cauchy-Riemann (CR) equation is shown on domains in Levi-flat CR manifolds for the class of C∞ forms up to the boundary. These domains are transversal intersections of the CR submanifold with pseudoconvex domains with piecewise smooth boundary.

On Cauchy-Riemann submanifolds whose local geodesic symmetries preserve the fundamental form

On Cauchy-Riemann submanifolds whose local geodesic symmetries preserve the fundamental form

Submanifolds of generalized Hopf manifolds, type numbers and the first Chern class of the normal bundle

Any orientable real hypersurface M of a complex Hopf manifold (carrying the locally conformal Kaehler (l.c.K.) metric discovered by I.Vaisman [33]) has a natural f-structure P as a generic Cauchy-Riemann submanifold; we show (cf. our § 5) that if P anti-commutes with the Weingarten operator, then the type number of the hypersurface is less equal than 1. Moreover, M carries the natural almost contact metrical structure observed by Y.Tashiro [30]; if its almost contact vector is an eigenvector of the Weingarten operator corresponding to a nowhere vanishing eigenfunction and the holomorphic distribution is involutive, then M is foliated with globally conformai Kaehler manifolds (cf. our § 5), provided that some restrictions on the type number of M are imposed. We derive (cf. our § 6) a «Simons type» formula and apply it to compact orientable hypersurfaces with non-negative sectional curvature (in a complex Hopf manifold) and parallel mean curvature vector. Several examples of submanifolds of l.c.K. manifolds are exhibited in § 3. Our § 7 studies complex submanifolds of generalized Hopf manifolds; for instance, we show that the first Chern class of the normal bundle of a complex submanifold having a flat normal connection is vanishing.

Cauchy-Riemann submanifolds of locally conformal Kaehler manifolds

Cauchy-Riemann submanifolds of locally conformal Kaehler manifolds