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Register a new userThe Leray transform: factorization, dual $CR$ structures and model hypersurfaces in $mathbb{C}mathbb{P}^2$
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We compute the exact norms of the Leray transforms for a family $mathcal{S}_{beta}$ of unbounded hypersurfaces in two complex dimensions. The $mathcal{S}_{beta}$ generalize the Heisenberg group, and provide local projective approximations to any smooth, strongly $mathbb{C}$-convex hypersurface $mathcal{S}_{beta}$ to two orders of tangency. This work is then examined in the context of projective dual $CR$-structures and the corresponding pair of canonical dual Hardy spaces associated to $mathcal{S}_{beta}$, leading to a universal description of the Leray transform and a factorization of the transform through orthogonal projection onto the conjugate dual Hardy space.
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The Leray transform and related boundary operators are studied for a class of convex Reinhardt domains in $mathbb C^2$. Our class is self-dual; it contains some domains with less than $C^2$-smooth boundary and also some domains with smooth boundary and degenerate Levi form. $L^2$-regularity is proved, and essential spectra are computed with respect to a family of boundary measures which includes surface measure. A duality principle is established providing explicit unitary equivalence between operators on domains in our class and operators on the corresponding polar domains. Many of these results are new even for the classical case of smoothly bounded strongly convex Reinhardt domains.
We show that the boundary of any bounded strongly pseudoconvex complete circular domain in $mathbb C^2$ must contain points that are exceptionally tangent to a projective image of the unit sphere.
Consider a $2$-nondegenerate constant Levi rank $1$ rigid $mathcal{C}^omega$ hypersurface $M^5 subset mathbb{C}^3$ in coordinates $(z, zeta, w = u + iv)$: [ u = Fbig(z,zeta,bar{z},bar{zeta}big). ] The Gaussier-Merker model $u=frac{zbar{z}+ frac{1}{2}z^2bar{zeta}+frac{1}{2} bar{z}^2 zeta}{1-zeta bar{zeta}}$ was shown by Fels-Kaup 2007 to be locally CR-equivalent to the light cone ${x_1^2+x_2^2-x_3^2=0}$. Another representation is the tube $u=frac{x^2}{1-y}$. Inspired by Alexander Isaev, we study rigid biholomorphisms: [ (z,zeta,w) longmapsto big( f(z,zeta), g(z,zeta), rho,w+h(z,zeta) big) =: (z,zeta,w). ] The G-M model has 7-dimensional rigid automorphisms group. A Cartan-type reduction to an e-structure was done by Foo-Merker-Ta in 1904.02562. Three relative invariants appeared: $V_0$, $I_0$ (primary) and $Q_0$ (derived). In Pocchiolas formalism, Section 8 provides a finalized expression for $Q_0$. The goal is to establish the Poincare-Moser complete normal form: [ u = frac{zbar{z}+frac{1}{2},z^2bar{zeta} +frac{1}{2},bar{z}^2zeta}{ 1-zetabar{zeta}} + sum_{a,b,c,d atop a+cgeqslant 3}, G_{a,b,c,d}, z^azeta^bbar{z}^cbar{zeta}^d, ] with $0 = G_{a,b,0,0} = G_{a,b,1,0} = G_{a,b,2,0}$ and $0 = G_{3,0,0,1} = {rm Im}, G_{3,0,1,1}$. We apply the method of Chen-Merker 1908.07867 to catch (relative) invariants at every point, not only at the central point, as the coefficients $G_{0,1,4,0}$, $G_{0, 2, 3, 0}$, ${rm Re} G_{3,0,1,1}$. With this, a brige Poincare $longleftrightarrow$ Cartan is constructed. In terms of $F$, the numerators of $V_0$, $I_0$, $Q_0$ incorporate 11, 52, 824 differential monomials.
We give a complete characterization of those disk bundles over surfaces which embed as rationally convex strictly pseudoconvex domains in $mathbb{C}^2$. We recall some classical obstructions and prove some deeper ones related to symplectic and contact topology. We explain the close connection to Lagrangian surfaces with isolated singularities and develop techniques for constructing such surfaces. Our proof also gives a complete characterization of Lagrangian surfaces with open Whitney umbrellas, answering a question first posed by Givental in 1986.
The goal of this article is to prove the Sum of Squares Conjecture for real polynomials $r(z,bar{z})$ on $mathbb{C}^3$ with diagonal coefficient matrix. This conjecture describes the possible values for the rank of $r(z,bar{z}) |z|^2$ under the hypothesis that $r(z,bar{z})|z|^2=|h(z)|^2$ for some holomorphic polynomial mapping $h$. Our approach is to connect this problem to the degree estimates problem for proper holomorphic monomial mappings from the unit ball in $mathbb{C}^2$ to the unit ball in $mathbb{C}^k$. DAngelo, Kos, and Riehl proved the sharp degree estimates theorem in this setting, and we give a new proof using techniques from commutative algebra. We then complete the proof of the Sum of Squares Conjecture in this case using similar algebraic techniques.
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