On the problem of deciding whether a holomorphic vector field is complete

López, Jorge; Muciño–Raymundo, Jesús

doi:10.1007/978-3-0348-8698-7_13

Cited by 7 publications

(5 citation statements)

References 18 publications

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“…Such a bundle can be identified with C 2 in our case. There exists a well-established theory about completeness of holomorphic vector fields X on C 2 in the case that they are polynomial (see the review [27] or more recent works such as [11,10]); such a theory includes links to the real analytic case (see for example [22]). The usage of complex variable allows to obtain very powerful results in comparison with those obtained just by means of the underlying real variables 4 .…”

Section: 4mentioning

confidence: 99%

The Ehlers-Kundt conjecture about gravitational waves and dynamical systems

Flores

Sánchez

2020

Journal of Differential Equations

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Ehlers-Kundt conjecture is a physical assertion about the fundamental role of plane waves for the description of gravitational waves. Mathematically, it becomes equivalent to a problem on the Euclidean plane R 2 with a very simple formulation in Classical Mechanics: given a non-necessarily autonomous potential V (z, u), (z, u) ∈ R 2 × R, harmonic in z (i.e. source-free), the trajectories of its associated dynamical systemz(s) = −∇zV (z(s), s) are complete (they live eternally) if and only if V (z, u) is a polynomial in z of degree at most 2 (so that V is a standard mathematical idealization of vacuum). Here, the conjecture is solved in the significative case that V is bounded polynomially in z for finite values of u ∈ R. The mathematical and physical implications of this polynomial EK conjecture, as well as the non-polynomial one, are discussed beyond their original scope.

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Section: 4mentioning

confidence: 99%

The Ehlers-Kundt conjecture about gravitational waves and dynamical systems

Flores

Sánchez

2020

Journal of Differential Equations

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“…(1) C, X = P (z) ∂ ∂z with P a polynomial of degree at most 1, (2) C, X = P (z) ∂ ∂z with P a polynomial of degree at most 2, (3) C * , X = λz ∂ ∂z , with λ ∈ C * , (4) M = C/Γ, X = λ ∂ ∂z , where M is a complex torus. For a proof see [47].…”

Section: Complex Flowsmentioning

confidence: 99%

On the geometry, flows and visualization of singular complex analytic vector fields on Riemann surfaces

Alvarez-Parrilla,

Muciño-Raymundo,

Solorza-Calderón

et al. 2018

Preprint

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Motivated by the wild behavior of isolated essential singularities in complex analysis, we study singular complex analytic vector fields X on arbitrary Riemann surfaces M . By vector field singularities we understand zeros, poles, isolated essential singularities and accumulation points of the above kind. In this framework, a singular analytic vector field X has canonically associated; a 1-form, a quadratic differential, a flat metric (with a geodesic foliation), a global distinguished parameter or C-flow box Ψ X , a Newton map Φ X , and a Riemann surface R X arising from the maximal C-flow of X. We show that every singular complex analytic vector field X on a Riemann surface is in fact both a global pullback of the constant vector field under Ψ X and of the radial vector field on the sphere under Φ X . As a result of independent interest, we show that the maximal analytic continuation of the a local C-flow of X is univalued on the Riemann surface R X ⊂ M × Ct, where R X is the graph of Ψ X . Furthermore we explore the geometry of singular complex analytic vector fields and present a geometrical method that enables us to obtain the solution, without numerical integration, to the differential equation that provides the C-flow of the vector field. We discuss the theory behind the method, its implementation, comparison with some integrationbased techniques, as well as examples of the visualization of complex vector fields on the plane, sphere and torus. Applications to visualization of complex valued functions is discussed including some advantages between other methods.

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“…This (C, +)-action is smooth on M * but generically is not well-defined for all complex time [19]; more precisely, it is discontinuous at the poles of the vector field. Hence, it is simpler for us to provide the infinitesimal description of these actions using F and G.…”

Section: Corollarymentioning

confidence: 99%

“…Further applications are to study real Hamiltonian vector fields having an additional first integral [25], to characterize complete holomorphic vector fields on higher dimensional complex manifolds [19], and to describe meromorphic quadratic differentials with prescribed singularities [5].…”

Section: Introductionmentioning

confidence: 99%

Complex structures adapted to smooth vector fields

Muciño–Raymundo¹

2002

Math Ann

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In this paper we study the flat geometry and real dynamics of meromorphic vector fields on compact Riemann surfaces. Necessary and sufficient conditions to assert the existence of meromorphic vector fields with prescribed singularities are given. A characterization of the real dynamics of meromorphic vector fields is also given. Several explicit examples of meromorphic vector fields, using singular flat metrics, are provided.

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On the problem of deciding whether a holomorphic vector field is complete

Cited by 7 publications

References 18 publications

The Ehlers-Kundt conjecture about gravitational waves and dynamical systems

The Ehlers-Kundt conjecture about gravitational waves and dynamical systems

On the geometry, flows and visualization of singular complex analytic vector fields on Riemann surfaces

Complex structures adapted to smooth vector fields

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