Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userProjective-umbilic points of circular real hypersurfaces in $mathbb C^2$
127
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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.
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 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 study the local equivalence problem for real-analytic ($mathcal{C}^omega$) hypersurfaces $M^5 subset mathbb{C}^3$ which, in coordinates $(z_1, z_2, w) in mathbb{C}^3$ with $w = u+i, v$, are rigid: [ u ,=, Fbig(z_1,z_2,overline{z}_1,overline{z}_2big), ] with $F$ independent of $v$. Specifically, we study the group ${sf Hol}_{sf rigid}(M)$ of rigid local biholomorphic transformations of the form: [ big(z_1,z_2,wbig) longmapsto Big( f_1(z_1,z_2), f_2(z_1,z_2), a,w + g(z_1,z_2) Big), ] where $a in mathbb{R} backslash {0}$ and $frac{D(f_1,f_2)}{D(z_1,z_2)} eq 0$, which preserve rigidity of hypersurfaces. After performing a Cartan-type reduction to an appropriate ${e}$-structure, we find exactly two primary invariants $I_0$ and $V_0$, which we express explicitly in terms of the $5$-jet of the graphing function $F$ of $M$. The identical vanishing $0 equiv I_0 big( J^5F big) equiv V_0 big( J^5F big)$ then provides a necessary and sufficient condition for $M$ to be locally rigidly-biholomorphic to the known model hypersurface: [ M_{sf LC} colon u ,=, frac{z_1,overline{z}_1 +frac{1}{2},z_1^2overline{z}_2 +frac{1}{2},overline{z}_1^2z_2}{ 1-z_2overline{z}_2}. ] We establish that $dim, {sf Hol}_{sf rigid} (M) leq 7 = dim, {sf Hol}_{sf rigid} big( M_{sf LC} big)$ always. If one of these two primary invariants $I_0 otequiv 0$ or $V_0 otequiv 0$ does not vanish identically, we show that this rigid equivalence problem between rigid hypersurfaces reduces to an equivalence problem for a certain $5$-dimensional ${e}$-structure on $M$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
