Singular Levi-flat hypersurfaces in complex projective space induced by curves in the Grassmannian

Let $H \subset {\mathbb P}^n$ be a real-analytic subvariety of codimension one induced by a real-analytic curve in the Grassmannian $G(n+1,n)$. Assuming $H$ has a global defining function, we prove $H$ is Levi-flat, the closure of its smooth points of top dimension is a union of complex hyperplanes, and its singular set is either of dimension $2n-2$ or dimension $2n-4$. If the singular set is of dimension $2n-4$, then we show the hypersurface is algebraic and the Levi-foliation extends to a singular holomorphic foliation of ${\mathbb P}^n$ with a meromorphic (rational of degree 1) first integral. In this case, $H$ is in some sense simply a complex cone over an algebraic curve in ${\mathbb P}^1$. Similarly if $H$ has a degenerate singularity, then $H$ is also algebraic. If the dimension of the singular set is $2n-2$ and is nondegenerate, we show by construction that the hypersurface need not be algebraic nor semialgebraic. We construct a Levi-flat real-analytic subvariety in ${\mathbb P}^2$ of real codimension 1 with compact leaves that is not contained in any proper real-algebraic subvariety of ${\mathbb P}^2$. Therefore a straightforward analogue of Chow's theorem for Levi-flat hypersurfaces does not hold.Comment: 13 pages, 1 figure, add missing hypotheses to first theorem, reorganized and added some detail

Extension of Levi-flat hypersurfaces past CR boundaries

Local conditions on boundaries of $C^\infty$ Levi-flat hypersurfaces, in case the boundary is a generic submanifold, are studied. For nontrivial real analytic boundaries we get an extension and uniqueness result, which forces the hypersurface to be real analytic. This allows us to classify all real analytic generic boundaries of Levi-flat hypersurfaces in terms of their normal coordinates. For the remaining case of generic real analytic boundary we get a weaker extension theorem. We find examples to show that these two extension results are optimal. Further, a class of nowhere minimal real analytic submanifolds is found, which is never the boundary of even a $C^2$ Levi-flat hypersurface.Comment: 15 pages; latex, amsrefs; fix statement and proof of theorem 3.1; accepted in Indiana Univ. Math.

Polynomials constant on a hyperplane and CR maps of spheres

We prove a sharp degree bound for polynomials constant on a hyperplane with a fixed number of nonnegative distinct monomials. This bound was conjectured by John P. D'Angelo, proved in two dimensions by D'Angelo, Kos and Riehl and in three dimensions by the authors. The current work builds upon these results to settle the conjecture in all dimensions. We also give a complete description of all polynomials in dimensions 4 and higher for which the sharp bound is obtained. The results prove the sharp degree bounds for monomial CR mappings of spheres in all dimensions.Comment: 17 pages, 10 figures; accepted to Illinois J. Math., added 3 figures and improved expositio