Formal Theory

2011 ◽  
Author(s):  
Charles Fefferman ◽  
C. Robin Graham
Keyword(s):  
Normal Form ◽  
Ricci Curvature ◽  
Infinite Order ◽  
Formal Theory ◽  
Formal Series ◽  
Ricci Flat ◽  
Convergence Results ◽  
Real Analytic

This chapter presents proof of Theorem 2.9 for n > 2. It further notes that similar arguments using the form of the perturbation formulae (3.32) for the Ricci curvature show that the metrics constructed in Theorems 3.7, 3.9 and 3.10 are the only formal expansions of metrics for ρ‎ > 0 or ρ‎ < 0 involving positive powers of ¦ ρ‎ r ρ‎ and log ¦ ρ‎ r ρ‎, which are homogeneous of degree 2, Ricci-flat to infinite order, and in normal form. Convergence of formal series determined by Fuchsian problems such as these in the case of real-analytic data has been considered by several authors. In particular, results of [BaoG] can be applied to establish the convergence of the series occurring in Theorems 3.7 and 3.9 (and also in Theorem 3.10 if the obstruction tensor vanishes) if g and h are real-analytic. Convergence results including also the case when log terms occur in Theorem 3.10 are contained in [K].

Ambient Metrics

2011 ◽  
Author(s):  
Charles Fefferman ◽  
C. Robin Graham
Keyword(s):  
Initial Value Problems ◽  
Infinite Order ◽  
Formal Theory ◽  
Formal Series ◽  
Precise Description ◽  
Initial Value ◽  
Real Analytic

This chapter presents the full infinite-order formal theory for ambient metric forms, including the freedom at order n/2 in all dimensions and the precise description of the log terms when n ≤ 4 is even. The description of the solutions with freedom at order n/2 and log terms extends and sharpens results of Kichenassamy [K]. Convergence of the formal series determined by singular nonlinear initial value problems of this type has been considered by several authors; these results imply that the formal series converge if the data are real-analytic.

scholarly journals Convergent normal form for real hypersurfaces at a generic Levi-degeneracy

2019 ◽  
Vol 2019 (749) ◽  
pp. 201-225
Author(s):  
Ilya Kossovskiy ◽  
Dmitri Zaitsev
Keyword(s):  
Normal Form ◽  
Moduli Space ◽  
Real Hypersurface ◽  
Convergence Result ◽  
Explicit Description ◽  
Real Hypersurfaces ◽  
Real Analytic ◽  
Real Analytic Hypersurfaces

Abstract We construct a complete convergent normal form for a real hypersurface in {\mathbb{C}^{N}} , {N\geq 2} , at a generic Levi-degeneracy. This seems to be the first convergent normal form for a Levi-degenerate hypersurface. As an application of the convergence result, we obtain an explicit description of the moduli space of germs of real-analytic hypersurfaces with a generic Levi-degeneracy. As another application, we obtain, in the spirit of the work of Chern and Moser [6], distinguished curves inside the Levi-degeneracy set that we call degenerate chains.

scholarly journals Convergence of the Chern–Moser–Beloshapka normal forms

2020 ◽  
Vol 2020 (765) ◽  
pp. 205-247
Author(s):  
Bernhard Lamel ◽  
Laurent Stolovitch
Keyword(s):  
Normal Form ◽  
Normal Forms ◽  
Direct Proof ◽  
Sufficient Condition ◽  
Normalizing Transformation ◽  
General Sufficient Condition ◽  
Real Analytic ◽  
Moser Normal Form

AbstractIn this article, we give a normal form for real-analytic, Levi-nondegenerate submanifolds of{\mathbb{C}^{N}}of codimension{d\geq 1}under the action of formal biholomorphisms. We find a very general sufficient condition on the formal normal form that ensures that the normalizing transformation to this normal form is holomorphic. In the case{d=1}our methods in particular allow us to obtain a new and direct proof of the convergence of the Chern–Moser normal form.

scholarly journals On holomorphic extension of functions on singular real hypersurfaces inℂn

2001 ◽  
Vol 26 (3) ◽  
pp. 173-178
Author(s):  
Tejinder S. Neelon
Keyword(s):  
Limit Point ◽  
Infinite Order ◽  
Holomorphic Extension ◽  
Real Hypersurfaces ◽  
Zero Set ◽  
Riemann Equation ◽  
Real Analytic ◽  
Cauchy Riemann ◽  
Extension Of Functions ◽  
Tangential Cauchy Riemann Equation

The holomorphic extension of functions defined on a class of real hypersurfaces inℂnwith singularities is investigated. Whenn=2, we prove the following: everyC1function onΣthat satisfies the tangential Cauchy-Riemann equation on boundary of{(z,w)∈ℂ2:|z|k<P(w)},P∈C1,P≥0andP≢0, extends holomorphically inside provided the zero setP(w)=0has a limit point orP(w)vanishes to infinite order. Furthermore, ifPis real analytic then the condition is also necessary.

Self-dual Poincaré Metrics

2011 ◽  
Author(s):  
Charles Fefferman ◽  
C. Robin Graham
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Existence And Uniqueness ◽  
Formal Theory ◽  
Einstein Metric ◽  
Real Analytic ◽  
Formal Power

As an application of the formal theory for Poincaré metrics, this chapter presents a formal power series proof of a result of LeBrun [LeB] asserting the existence and uniqueness of a real-analytic self-dual Einstein metric in dimension 4 defined near the boundary with prescribed real-analytic conformal infinity.

INFINITE ORDER PARAMETRIC NORMAL FORM OF HOPF SINGULARITY

2008 ◽  
Vol 18 (11) ◽  
pp. 3393-3408 ◽  
Author(s):  
MAJID GAZOR ◽  
PEI YU
Keyword(s):  
Normal Form ◽  
State Space ◽  
Integral Domain ◽  
Infinite Order ◽  
State Parameter ◽  
Efficient Computation ◽  
Graded Ring ◽  
Codimension One ◽  
Lie Bracket ◽  
Graded Module

In this paper, we introduce a suitable algebraic structure for efficient computation of the parametric normal form of Hopf singularity based on a notion of formal decompositions. Our parametric state and time spaces are respectively graded parametric Lie algebra and graded ring. As a consequence, the parametric state space is also a graded module. Parameter space is observed as an integral domain as well as a vector space, while the near-identity parameter map acts on the parametric state space. The method of multiple Lie bracket is used to obtain an infinite order parametric normal form of codimension-one Hopf singularity. Filtration topology is revisited and proved that state, parameter and time (near-identity) maps are continuous. Furthermore, parametric normal form is a convergent process with respect to filtration topology. All the results presented in this paper are verified by using Maple.

scholarly journals Fixed points of elliptic reversible transformations with integrals

1996 ◽  
Vol 16 (4) ◽  
pp. 683-702
Author(s):  
Xianghong Gong
Keyword(s):  
Normal Form ◽  
Fixed Points ◽  
Periodic Points ◽  
General Transformation ◽  
Reversible Transformation ◽  
Level Curves ◽  
Hyperbolic Complex ◽  
Holomorphic Transformation ◽  
Real Analytic ◽  
Real Hyperplane

AbstractWe show that for a certain family of integrable reversible transformations, the curves of periodic points of a general transformation cross the level curves of its integrals. This leads to the divergence of the normal form for a general reversible transformation with integrals. We also study the integrable holomorphic reversible transformations coming from real analytic surfaces in ℂ2 with non-degenerate complex tangents. We show the existence of real analytic surfaces with hyperbolic complex tangents, which are contained in a real hyperplane, but cannot be transformed into the Moser—Webster normal form through any holomorphic transformation.

NEW STUDY ON THE CENTER PROBLEM AND BIFURCATIONS OF LIMIT CYCLES FOR THE LYAPUNOV SYSTEM (II)

2009 ◽  
Vol 19 (09) ◽  
pp. 3087-3099 ◽  
Author(s):  
YIRONG LIU ◽  
JIBIN LI
Keyword(s):  
Normal Form ◽  
Limit Cycles ◽  
Formal Series ◽  
Center Problem ◽  
Lyapunov System ◽  
Cubic Systems ◽  
Lyapunov Constants

The center problem and bifurcations of limit cycles for the Lyapunov system are continuously studied. We shall prove that we can construct successively a formal series such that the Lyapunov system is reduced a half-normal form. From the coefficients of the half-normal form, we obtain directly the Lyapunov constants of the origin. As examples, for two classes of cubic systems, the center and focus problem, and multiple bifurcations of limit cycles are studied.

scholarly journals Four-dimensional Einstein manifolds with Heisenberg symmetry

2021 ◽  
Author(s):  
V. Cortés ◽  
A. Saha
Keyword(s):  
Heisenberg Group ◽  
Ricci Curvature ◽  
Einstein Metrics ◽  
Hyperbolic Metric ◽  
Einstein Manifolds ◽  
Cohomogeneity One ◽  
Ricci Flat ◽  
Dimensional Group ◽  
The One ◽  
Group Of Isometries

AbstractWe classify Einstein metrics on $$\mathbb {R}^4$$ R 4 invariant under a four-dimensional group of isometries including a principal action of the Heisenberg group. We consider metrics which are either Ricci-flat or of negative Ricci curvature. We show that all of the Ricci-flat metrics, including the simplest ones which are hyper-Kähler, are incomplete. By contrast, those of negative Ricci curvature contain precisely two complete examples: the complex hyperbolic metric and a metric of cohomogeneity one known as the one-loop deformed universal hypermultiplet.

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