Singular Levi-flat hypersurfaces

Abstract

In this thesis we study germs of singular real-analytic Levi-flat hypersurfaces with two distinct purposes. We show the existence of normal forms for germs of singular Levi-flat hypersurfaces which are defined by the vanishing of the real part of complex quasihomogeneous polynomials with isolated singularity. This result generalizes previous results of Burns-Gong [7] and Fernández-Pérez [15]. Furthermore, we show the existence of two new rigid normal forms for germs of singularreal-analytic Levi-flat hypersurfaces which are preserved by a change of isochore coordinates, that is, a change of coordinates that preserves volume. Moreover, we address the problem of finding sufficient conditions to guarantee the coincidence of the level sets of a holomorphic function with the leaves of the Levi foliation on a germ of a real-analytic Levi-flat hypersurface with isolated singularity. For a germ of irreducible real-analytic Levi-flat hypersurface at (...) with a nondicritical isolated singularity, we show that the leaves of the Levi foliation coincide with the level sets of real values of a holomorphic function. In the dicritical case, acounter-example of this result is given.