On Convergent Poincaré-Moser Reduction for Levi Degenerate Embedded $5$-Dimensional CR Manifolds

Wei Guo Foo; Joel Merker; The-Anh Ta

Home/Archive/arXiv

arXiv · 2003.01952

On Convergent Poincaré-Moser Reduction for Levi Degenerate Embedded $5$-Dimensional CR Manifolds

Wei Guo Foo·Joel Merker·The-Anh Ta

View original source ↗Read source document ↗Export metadata ↓

Abstract

Applying Lie's theory, we show that any $\mathcal{C}^ω$ hypersurface $M^5 \subset \mathbb{C}^3$ in the class $\mathfrak{C}_{2,1}$ carries Cartan-Moser chains of orders $1$ and $2$. Integrating and straightening any order $2$ chain at any point $p \in M$ to be the $v$-axis in coordinates $(z, ζ, w = u + i\, v)$ centered at $p$, we show that there exists a (unique up to 5 parameters) convergent change of complex coordinates fixing the origin in which $γ$ is the $v$-axis so that $M = \{u=F(z,ζ,\overline{z},\overlineζ,v)\}$ has Poincaré-Moser reduced equation: \begin{align} u & = z\overline{z} + \tfrac{1}{2}\,\overline{z}^2ζ+ \tfrac{1}{2}\,z^2\overlineζ + z\overline{z}ζ\overlineζ + \tfrac{1}{2}\,\overline{z}^2ζζ\overlineζ + \tfrac{1}{2}\,z^2\overlineζζ\overlineζ + z\overline{z}ζ\overlineζζ\overlineζ \\ & + 2{\rm Re} \{ z^3\overlineζ^2 F_{3,0,0,2}(v) + ζ\overlineζ ( 3\,{z}^2\overline{z}\overlineζ F_{3,0,0,2}(v) ) \} \\ & + 2{\rm Re} \{ z^5\overlineζ F_{5,0,0,1}(v) + z^4\overlineζ^2 F_{4,0,0,2}(v) + z^3\overline{z}^2\overlineζ F_{3,0,2,1}(v) + z^3\overline{z}\overlineζ^2 F_{3,0,1,2}(v) + z^3{\overlineζ}^3 F_{3,0,0,3}(v) \} \\ & + z^3\overline{z}^3 {\rm O}_{z,\overline{z}}(1) + 2{\rm Re} ( \overline{z}^3ζ{\rm O}_{z,ζ,\overline{z}}(3) ) + ζ\overlineζ\, {\rm O}_{z,ζ,\overline{z},\overlineζ}(5). \end{align} The values at the origin of Pocchiola's two primary invariants are: \[ W_0 = 4\overline{F_{3,0,0,2}(0)}, \quad\quad J_0 = 20\, F_{5,0,0,1}(0). \] The proofs are detailed, accessible to non-experts. The computer-generated aspects (upcoming) have been reduced to a minimum.

Explore related subjects

math.CV math.DG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Wei Guo Foo, Joel Merker, The-Anh Ta. 2021-12-16. On Convergent Poincaré-Moser Reduction for Levi Degenerate Embedded $5$-Dimensional CR Manifolds. https://arxiv.org/abs/2003.01952

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Product Principle for Harmonic Schwarz Lemmas: Boxes, Polydiscs, and Metric Geometry

We establish an exact product principle for the Euclidean operator norm of differentials of harmonic maps. For a bounded domain $G\subset\R^m$ and $p\in G$, let $M_G(p)$ denote the supremum of $\|dF_0\|$ over harmonic maps $F:\D\to G$ with $F(0)=p$. For bounded domains $G_j\subset\R^{m_j}$, we prove \[ M_{G_1\times\cdots\times G_N}(p_1,\ldots,p_N)^2 =\sum_{j=1}^N M_{G_j}(p_j)^2. \] The theorem separates the geometry of the individual factors from the Euclidean geometry of the product: the factor extremal constants combine by a sum-of-squares law, while equality is governed by a single compatibility condition, namely a common maximizing direction for the component differentials. Neither convexity nor attainment of the factor suprema is required. Combining the product principle with the sharp interval and disk factor problems yields exact operator-norm estimates and all equality cases for harmonic maps into boxes and polydiscs. In both families, for every extremal map, the real differential at the origin has one-dimensional image. The same factor constants also define coordinatewise metrics for which the harmonic contraction estimate is sharp when the source disk is equipped with its Poincaré metric of curvature $-1$. For boxes, the resulting metric is complete and equals twice the restriction of the Kobayashi-Royden metric of the product of vertical strips. For polydiscs, the harmonic product metric is pointwise maximal among contracting metrics of the form $\max_j a_j(p)|v_j|$, with $a_j(p)>0$. It is strictly smaller than twice the Kobayashi-Royden metric on every nonzero tangent vector, and its induced path metric is incomplete.

math.CV↗

arXiv2026

bi-Lipschitz versus Analytic equivalence of two variable complex quasihomogeneous function-germs

In this paper we address the problem of classifying complex (non-homogeneous) quasihomogeneous polynomials in two variables under bi-Lipschitz equivalence. We prove that pairs of such polynomials are (right) bi-Lipschitz equivalent as function-germs at $0\in\mathbb{C}^{2}$ iff they are analytically equivalent.

math.CV↗

arXiv2026

Convolution Regularization Preserves the $L^2$-Estimate Property for $(1,n)$-Forms

In this paper, we prove that the $L^2$-estimate property for $(1,n)$-forms is preserved under the standard convolution regularization. As applications, we show that any singular Hermitian metric satisfying the optimal or multiple coarse $L^2$-estimate property for $(n,1)$ or $(1,n)$-forms is Griffiths semi-positive. This resolves a question posed by Deng--Ning--Wang and a question by Inayama.

math.CV↗