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Register a new userHolomorphic immersions of bi-disks into $9$ dimensional real hypersurfaces with Levi signature $(2, 2)$
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Wei Guo Foo
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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.
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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 $M$.
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