اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدHolomorphic immersions of bi-disks into $9$ dimensional real hypersurfaces with Levi signature $(2, 2)$
234
0
0.0
(
0
)
تأليف
Wei Guo Foo
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Inspired by an article of R. Bryant on holomorphic immersions of unit disks into Lorentzian CR manifolds, we discuss the application of Cartans method to the question of the existence of bi-disk $mathbb{D}^{2}$ in a smooth $9$-dimensional real analytic real hypersurface $M^{9}subsetmathbb{C}^{5}$ with Levi signature $(2,2)$ passing through a fixed point. The result is that the lift to $M^{9}times U(2)$ of the image of the bi-disk in $M^{9}$ must lie in the zero set of two complex-valued functions in $M^{9}times U(2)$. We then provide an example where one of the functions does not identically vanish, thus obstructing holomorphic immersions.
قيم البحث
اقرأ أيضاً
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}_2bi
g), ] 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 $M$.
Using a bigraded differential complex depending on the CR and pseudohermitian structure, we give a characterization of three-dimensional strongly pseudoconvex pseudo-hermitian CR-manifolds isometrically immersed in Euclidean space $mathbb{R}^n$ in te
rms of an integral representation of Weierstrass type. Restricting to the case of immersions in $mathbb{R}^4$, we study harmonicity conditions for such immersions and give a complete classification of CR-pluriharmonic immersions.
This is the second part of a series of two papers dedicated to a systematic study of holomorphic Jacobi structures. In the first part, we introduced and study the concept of a holomorphic Jacobi manifold in a very natural way as well as various tools
. In the present paper, we solve the integration problem for holomorphic Jacobi manifolds by proving that they integrate to complex contact groupoids. A crucial tool in our proof is what we call the homogenization scheme, which allows us to identify holomorphic Jacobi manifolds with homogeneous holomorphic Poisson manifolds and holomorphic contact groupoids with homogeneous complex symplectic groupoids.
We show that, for a closed orientable n-manifold, with n not congruent to 3 modulo 4, the existence of a CR-regular embedding into complex (n-1)-space ensures the existence of a totally real embedding into complex n-space. This implies that a closed
orientable (4k+1)-manifold with non-vanishing Kervaire semi-characteristic possesses no CR-regular embedding into complex 4k-space. We also pay special attention to the cases of CR-regular embeddings of spheres and of simply-connected 5-manifolds.
In this paper, we develop holomorphic Jacobi structures. Holomorphic Jacobi manifolds are in one-to-one correspondence with certain homogeneous holomorphic Poisson manifolds. Furthermore, holomorphic Poisson manifolds can be looked at as special case
s of holomorphic Jacobi manifolds. We show that holomorphic Jacobi structures yield a much richer framework than that of holomorphic Poisson structures. We also discuss the relationship between holomorphic Jacobi structures, generalized contact bundles and Jacobi-Nijenhuis structures.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
