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

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

Codimension two CR singular submanifolds and extensions of CR functions

Let $M \subset {\mathbb{C}}^{n+1}$, $n \geq 2$, be a real codimension two CR singular real-analytic submanifold that is nondegenerate and holomorphically flat. We prove that every real-analytic function on $M$ that is CR outside the CR singularities extends to a holomorphic function in a neighborhood of $M$. Our motivation is to prove the following analogue of the Hartogs-Bochner theorem. Let $\Omega \subset {\mathbb{C}}^n \times {\mathbb{R}}$, $n \geq 2$, be a bounded domain with a connected real-analytic boundary such that $\partial \Omega$ has only nondegenerate CR singularities. We prove that if $f \colon \partial \Omega \to {\mathbb{C}}$ is a real-analytic function that is CR at CR points of $\partial \Omega$, then $f$ extends to a holomorphic function on a neighborhood of $\overline{\Omega}$ in ${\mathbb{C}}^n \times {\mathbb{C}}$.Comment: 16 pages, 1 figure, fixed typos, updated references. To appear in Journal of Geometric Analysi

On Lewy extension for smooth hypersurfaces in Cn×R{\mathbb C}^n \times {\mathbb R}

We prove an analogue of the Lewy extension theorem for a real dimension $2n$ smooth submanifold $M \subset {\mathbb C}^{n}\times {\mathbb R}$, $n \geq 2$. A theorem of Hill and Taiani implies that if $M$ is CR and the Levi-form has a positive eigenvalue restricted to the leaves of ${\mathbb C}^n \times {\mathbb R}$, then every smooth CR function $f$ extends smoothly as a CR function to one side of $M$. If the Levi-form has eigenvalues of both signs, then $f$ extends to a neighborhood of $M$. Our main result concerns CR singular manifolds with a nondegenerate quadratic part $Q$. A smooth CR $f$ extends to one side if the Hermitian part of $Q$ has at least two positive eigenvalues, and $f$ extends to the other side if the form has at least two negative eigenvalues. We provide examples to show that at least two nonzero eigenvalues in the direction of the extension are needed.Comment: 22 pages; new examples and minor typo fixe

Initial monomial invariants of holomorphic maps

We study a new biholomorphic invariant of holomorphic maps between domains in different dimensions based on generic initial ideals. We start with the standard generic monomial ideals to find invariants for rational maps of spheres and hyperquadrics, giving a readily computable invariant in this important case. For example, the generic initial monomials distinguish all four inequivalent rational proper maps from the two to the three dimensional ball. Next, we associate to each subspace $X \subset {\mathcal O}(U)$ a generic initial monomial subspace, which is invariant under biholomorphic transformations and multiplication by nonzero functions. The generic initial monomial subspace is a biholomorphic invariant for holomorphic maps if the target automorphism is linear fractional as in the case of automorphisms of spheres or hyperquadrics.Comment: 16 pages, fixed typos, accepted to Math.