arXiv subjects
Publications and source records attributed to Claudio Meneghini.
We extend Painlevé's determinateness theorem to the case of first order ordinary differential equations in the complex domain with known terms allowed be multivalued in the dependent variable as well; multivaluedness is supposed to be resolved by proper coverings.
This thesis is concerned with extending the idea of geodesic completeness from pseudo-Riemannian to complex geometry: we take, however a completely holomorphicpoint of view; that is to say, a 'metric' will be a (meromorphic) symmetric section of the twice covariant holomorphic tensor bundle. We shall hint at the need of reformulating some aspects of the theory of differential equations in the complex domain, originating in the interpenetration betwixt differential and algebraic aspects when analytical continuation is pursued. The notion itself of path should be reformulated: we remark that geodesics will be defined on Riemann surfaces which are domains over regions in the complex plane. Of course geodesics will be eventually defined to be auto-parallel paths, but we shall focus our attention on the fact that the Levi-Civita connection will be meromorphic if the metric from which it is induced is allowed to have poles or even simply to lower somewhere in its rank. We shall study rather more deeply a class of manifolds, namely warped products of Riemann surfaces; some hypotheses concerning their metrics will be done, (metrics will be suppose to be 'coercive', in a sense that will be defined) but we shall show that the range of applicability of the yielded completeness theorems will not be exceedingly restricted.
In this paper we examinate some phenomena arising when a holomorphic germ is analytically continued.
We show, in an elementary way, that the Julia set of one-complex-variable entire functions is nonempty and perfect.
We exhibit an example of covering surface of C*, arising from analytical continuation of a holomorphic germ and failing to be a topological covering, due to singular points and regular ones being projected over the same slits in C*.
We propose to apply the idea of analytical continuation in the complex domain to the problem of geodesic completeness. We shall analyse rather in detail the cases of analytical warped products of real lines, these ones in parallel with their complex counterparts, and of Clifton-Pohl torus, to show that our definition sheds a bit of new light on the behaviour of 'singularities' of geodesics in space-time. We also show that some geodesics, which 'end' at finite time in the classical sense, can be naturally continued besides their ends. As a matter of fact, complex metrics naturally show a meromorphic behaviour, or a degenerating one, so we shall study also this fact in detail.
Nous montrons la densite des cycles repulsifs dans l'ensemble de Julia des fonctions meromorphes transcendentes a une variable complexe, sans utiliser le theoreme des cinq iles d'Ahlfors ni la theorie de Nevanlinna. ----- We prove that repelling cycles are dense in the Julia set of one-variable transcendental meromorphic functions, making use nor of Ahlfors' five-island theorem, nor of Nevanlinna's theory
We show that a natural complexification and a mild generalization of the idea of completeness guarantee geodesic completeness of Clifton-Pohl torus; we explicitely compute all of its geodesics.
After proving a multi-dimensional extension of Zalcman's renormalization lemma and considering maximality problems about dimensions, we find renormalizing polynomial families for iterated elementary mappings, extending this result to some kinds of correspondences (by means of 'algebraic' renormalizing families) and to the family of the iterated mappings of an automorphism of $\CI^2$ admitting a repulsive fixed point (by means of a family of polunomial automorphisms composed with a Fatou-Bieberbach one). All families will allow maximal-dimension renormalizations.
We propose, within the context of the dynamics of a holomorphic germ in CI^N, a definition of 'attracting basin' of a fixed point. We prove that the inverse germ of an endomorphism of CI^N with a repulsive fixed point in 0, satisfying a supplementary technical hypothesis, admits an attracting basin which could be described as a Fatou-Bieberbach domain 'with many leaves'.
We propose a new definition of geodesic completeness, based on analytical continuation in the complex domain: we apply this idea to Clifton-Pohl torus, relating, for each geodesic, completeness to the value of a function of initial conditions, called 'impulse'.
We extend Painlevé's determinateness theorem from the theory of ordinary differential equations in the complex domain allowing more general 'multiple-valued' Cauchy's problems. We study $C^0-$continuability (near singularities) of solutions.
In this paper we investigate possible extensions of the idea of geodesic completeness in complex manifolds, following two directions: metrics are somewhere allowed not to be of maximum rank, or to have 'poles' somewhere else. Geodesics are eventually defined on Riemann surfaces over regions in the Riemann sphere. Completeness theorems are given in the framework of warped products of Riemann surfaces.