I: Lie symmetries of partial differential equations and CR geometry

Joel Merker

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I: Lie symmetries of partial differential equations and CR geometry

Joel Merker

Complex Variables · Submitted 5 Mar 2007

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim The tube over the light cone in C^3 is shown to be the unique model (up to biholomorphisms) having CR automorphism group of maximal dimension equal to 10.

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What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.10 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.96 · compares with prior results 0.15 · mentions code or data availability 0.01.

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A general theory of rigid completely integrable analytic partial differential equations is endeavoured. The tube over the light cone in C^3 is shown to be the unique model (up to biholomorphisms) having CR automorphism group of maximal dimension equal to 10. Explicit formulas for the Lie prolongation of vector fields to jets spaces are derived.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	A general theory of rigid completely integrable analytic partial differential equations is endeavoured.	contribution 0.91	0.71	0.08	unclear 0.51
s1	The tube over the light cone in C^3 is shown to be the unique model (up to biholomorphisms) having CR automorphism group of maximal dimension equal to 10.	result 1.00	0.85	0.54	asserted 1.00
s2	Explicit formulas for the Lie prolongation of vector fields to jets spaces are derived.	contribution 0.51	0.91	0.08	asserted 1.00

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Original abstract

A general theory of rigid completely integrable analytic partial differential equations is endeavoured. The tube over the light cone in C^3 is shown to be the unique model (up to biholomorphisms) having CR automorphism group of maximal dimension equal to 10. Explicit formulas for the Lie prolongation of vector fields to jets spaces are derived.

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2 arXiv papers cite this in 17 sentences

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For this, we apply the techniques of computational differential algebra developed in [ref] which enable us here to explicitly execute the two-ways transfer between algebraic expressions in the jet of $Φ$ and algebraic expressions in the jet of $Θ$.

Nonrigid spherical real analytic hypersurfaces in C^2 · 1. Introduction · 1.00
Fortunately, here similarly as in [ref] , we may apply the techniques of computational differential algebra sketched in [ref] in order to explicitly express any algebraic expressions in the second-order jet of the $Φ_ k_1, k_2$ in terms of the fourth-order jet of $Θ$, and the appropriate general equation which we shall need: [equation] will be obtained in Section 4 below, after rather lengthy but elementary calculations, parts of which are inspired from [ref] .

Vanishing Hachtroudi curvature and local equivalence to the Heisenberg sphere · 1. Introduction · 0.97

Cites it as background 2 papers

At least since the publication in 1888 by Lie and Engel in Leipzig of the Theorie der Transformationsguppen, it is known in a very general context — see Chapter 10 of [ref] and also [ref] — that, to the whole family of Segre varieties: [equation] parametrized by the $n+1$ antiholomorphic variables $( z_1, …, z_n, w )$, one may canonically associate a completely integrable second-order system of partial differential equations whose general solution is precisely the function $Θ( z, z, w )$.

Vanishing Hachtroudi curvature and local equivalence to the Heisenberg sphere · Associated system of partial differential equations · 0.86
For a presentation of these concepts, the reader is referred to the extensive introductions of [ref] and also to [ref] for more about why dealing only with complex defining equations is natural and unavoidable when one wants to insert CR geometry in the wider universe of completely integrable systems of real or complex analytic partial differential equations. ,

Nonrigid spherical real analytic hypersurfaces in C^2 · 1. Introduction · 0.99

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